International Library of Psychology
Philosophy and Scientific Method
THE LOGICAL SYNTAX
OF LANGUAGE
International Library of Psychology
Philosophy and Scientific Method
by
GENERAL EDITOR: C. K. OGDEN, M.A,
Philosophical Studies .
The Misuse of Mind
Conflict and Dream* .
Tractatus Logico-Philosophicus .
Psychological Types* .
Scientific Thought* .
The Meaning of Meaning .
Individual Psychology
Speculations {Preface by Jacob Epstein)
The Psychology of Reasoning* .
The Philosophy of “As If” .
The Nature of Intelligence
Telepathy and Clairvoyance
The Growth of the Mind .
The Mentality of Apes
Psychology of Religious Mysticism
The Philosophy of Music .
The Psychology of a Musical Prodigy
Principles of Literary Criticism .
Metaphysical Foundation of Sciences
Thought and the Brain*
Physique and Character*
Psychology of Emotion
Problems of Personality
The Hls^ ory of Materialism
Personality* ....
Educational Psychology
Language and Thought of the Child
Sex and Repression in Savage Society*
Comparative Philosophy
Social Life hi the Animal World
H ow Animals Find their Way About'
The Social Insects
'I’heoretical Biology .
Possibility
The Technique of Controversy .
The Symbolic Process .
Political Pluralism ...
History of Chinese Political Thought
Integrai ive Psychology*
The Analysis of Matter
Plato’s Theory of Ethics , . ^
Historical Introduction to Modern Psychoj
Creative Imagination ....
Colour and Colour Theories
Biological Principles . . . . •
The Trauma of Birth . . .
The Statistical Method in Economics .
The Art of Interrogation ...
The Growth of Reason
Human Speech .....
Foundations of- Geometry and Induction
The Law s of Feeling ....
The Mental Development of the Child
Eidetic Imagery
The Concentric Meihod
The Foundations of Mathematics
The Philosophy of the Unconscious
Outlines of Greek Philosophy
The Psychology of Children's Drawings
Invention and the Unconscious .
The Theory of Legislation .
The Social Life of Monkeys
The Development of the Sexual Impulses
C oNS'i I ruTioN Types in Delinquency
The Sciences of Man in the Making
Ethical Relativity ....
{Magdalene CoUeget Cambridge)
by G. E. Moore, Litt.D.
by Karin Stephen
by W. H. R. Rivers, F.R.S.
by L. Wittgenstein
C. K
by
by
by C. G. Jung, M.D.
_ C. D. Broad, Litt.D.
Ogden and 1. A. Richards
by Alfred Adler
by T. E. Hulme
by Eugenio Rignano
by H. Vaihinger
^ L. L. Thurstone
. by R. Tischner
. by K. Koffka
. by W. Kohler
. J. H. Leuba
. 6v W. Pole, F.R.S.
by G. Revesz
by I. A. Richards
i>Y E. A. Burtf, Ph.D.
by H. PifiRON
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. by J. T. MacCurdy, M.D.
in honour of Morton Prince
by F. A. Lange
by R. G. Gordon, M.D.
. by Charles Fox
. . by J. Piaget
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. by Scott Buchanan
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by Bertrand Russell, F.R.S.
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by R. E. Money-Kyrle
. fry W. A. Willemse
fry £. A. Kirkpatrick
fry E. A. Westermarck
^ f
Cy ]
• Asterisks denote that other books by the same author are included in the Series.
THE LOGICAL SYNTAX
OF LANGUAGE
B/
RUDOLF CARNAP
Professorof Philosophy in the
Universityof Chicago
LONDON
ROUTLEDGE & KEGAN PAUL LTD
BROADWAY HOUSE; 68-74 CARTER LANE, E.C.4
First published in England igig
Translated by
AMETHE SMEATON (Countess von Zeppelin)
REPRINTED BY LITHOGRAPHY IN GREAT BRITAIN
BY JARROLD AND SONS, LIMITED, NORWICH
CONTENTS
PAGE
Preface to the English Edition xi
Foreword xiii
Introduction
§ I. What is Logical Syntax? i
§ 2. Languages as Calculi 4
PART 1 . THE DEFINITE LANGUAGE I
A. Rules of Formation for Language I
§3. Predicates and Functors ii
§ 4. Syntactical Gothic Symbols * * 5
§ 5. The Junction Symbols 18
§ 6. Universal and Existential Sentences . ^ . . 20
§ 7. The K-Operator 22
§ 8, The Definitions 23
§9. Sentences and Numerical Expressions . . . 25
B. Rules of Transformation for Language I
§ 10. General Remarks Concerning Transformation
Rules 27
§11. The Primitive Sentences of Language I . . 29
§ 12. The Rules of Inference of Language I . . 32
§ 13. Derivations and Proofs in Language I . . . 33
§ 14. Rules of Consequence for Language I . . 37
C. Remarks on the Definite Form of Language
§ 15. Definite and Indefinite 44
? 16. On Intuitionism 46
§ 16 a. Identity 49
§ 17. The Principle of Tolerance in Syntax . . 51
PART II. THE FORMAL CONSTRUCTION OF
THE SYNTAX OF LANGUAGE I
§18. The Syntax of I can be Formulated in I . . 53
§19. The Arithmetization of Syntax .... 54
§ 20. General Terms 58
Vi
CONTENTS
PAGE
§21. Rules of Formation : i . Numerical Expressions and
Sentences 62
§ 22. Rules of Formation : 2. Definitions ... 66
§ 23. Rules of Transformation 73
§ 24. Descriptive Syntax 76
§25. Arithmetical, Axiomatic, and Physical Syntax . 78
PART III. THE INDEFINITE LANGUAGE II
A. Rules of Formation for Language II
§ 26. The Symbolic Apparatus of Language II . 83
§ 27. The Classification of Types 84
§ 28*. Formation Rules for Numerical Expressions and
Sentences 87
§ 29. Formation Rules for Definitions .... 88
B. Rules of Transformation for Language II
§ 30. The Primitive Sentences of Language II . 90
§ 31. The Rules of Inference of Language II . . 94
§ 32. Derivations and Proofs in Language II . 95
§ 33. Comparison of the Primitive Sentences and Rules
of II with those of other Systems ... 96
C. Rules of Consequence for Language II
§ 34 a. Incomplete and Complete Criteria of Validity . 98
§34^. Reduction 102
§34^. Evaluation 106
§ 34 d. Definition of ‘ Analytic in II ’ and ‘ Contradictory
in II* no
§ 34 e. On Analytic and Contradictory Sentences of
Language II 115
§34/. Consequence in Language II .117
§ 34g. Logical C^ontent 126
§34/1. The Principles of Induction and Selection are
Analytic 12 1
§341. Language II is Non-Contradictpry . . . 124
CONTENTS
vii
PAGE
§35. Syntactical Sentences which Refer to Themselves 129
§36. Irresoluble Sentences 13 1
D. Further Development of Language II
§37. Predicates as Class-Symbols . .134
§ 38. The Elimination of Classes 136
§38^. On Existence Assumptions in Logic . 140
§386. Cardinal Numbers ... 142
§38c. Descriptious 1^4
§39. Real Numbers 14^
§ 40. The Language of Physics 149
PART IV. GENERAL SYNTAX V ' '
A. Object-Language and Syntax-Language
§41. On Syntactical Designations . . . 153
§ 42. On the Necessity of Distinguishing between an
Expression and its Designation . . . * . 156
§ 43. On the Admissibility of Indefinite Terms 160
§ 44. On the Admissibility of Impredicative Terms . 162
§45. Indefinite Terms in Syntax 165
B. The Syntax of any Language
(«) General Considerations
§46. Formation Rules 167
§47. Transformation Rules; d-Terms . . . 170
§48. c-Terms 172
§49. Content 175
§ 50. Logical and Descriptive Expressions ; Sub-Lan-
guage 177
§ 51. Logical and Physical Rules ... 180
§52. L-Terms; ‘Analytic’ and ‘Contradictory * . 182
(b) Variables
§ S 3 - Systems of Levels; Predicates and Functors . 186
§54. Substitution; Variables and Constants . . . 189
§55. Universal and Existential Operators . . .196
§ 56. Range 199
§ 57. Sentential Junctions .... 200
Vlll
CONTENTS
PAGE
(c) Arithmetic; NoR-Contradictoriness; the Antinomies
§58. Arithmetic 205
§ 59. The Non-Contradictoriness and Completeness of
a Language 207
§60 a. The Antinomies 21 1
§ 606. The Concepts ‘True' and ‘False* . 214
§6oc. The Syntactical Antinomies . . .217
§6od. Every Arithmetic is Defective .... 220
(d) Translation and Interpretation
§61. Translation from One Language into Another 222
§ 62. The Interpretation of a Language . 227
(e) Extensionality
§ 63. Quasi-Syntactical Sentences ....
§64. The Two Interpretations of Quasi-Syntactical
Sentences
§ 65. Extensionality in Relation to Partial Sentences .
§ 66. Extensionality in Relation to Partial Expressions .
§ 67. The Thesis of Extensionality . , . ,
§ 68. Intensional Sentences of the Autonymous Mode
of Speech
§ 69. Intensional Sentences of the Logic of Modalities .
§ 70. The Quasi-Syntactical and the Syntactical Methods
in the Logic of Modalities
§ 71. Is an Intensional Logic necessary?
(/) Relational Theory and Axiomatics
§ 71 a. Relational Theory 260
§ 71 b. Syntactical Terms of Relational Theory . 262
§7ic. Isomorphism 264
§ 71 d. The Non-Denumerable Cardinal Nutnbers . 267
§ 71 e. The Axiomatic Method 271
233
237
240
243
24s
247
250
256
257
CONTENTS
IX
PART V. PHILOSOPHY AND SYNTAX
A. On the Form of the Sentences Belonging to the
Logic of Science
§ 72. Philosophy Replaced by the Logic of Science
§ 73 . The Logic of Science is the Syntax of the Language
of Science
§ 74. Pseudo-Object-Sentences
§ 75. Sentences about Meaning
§ 76. Universal Words
§ 77. Universal Words in the Material Mode of Speech
§ 78. Confusion in Philosophy Caused by the Material
Mode of Speech
§ 79. Philosophical Sentences in the Material and in the
Formal Mode of Speech
§ 80. The Dangers of the Material Mode of Speech .
§81. The Admissibility of the Material Mode of Speech
B. The Logic of Science as Syntax
§ 82. The Physical Language
§83. The so-called Foundations of the Sciences .
§ 84. The Problem of the Foundation of Mathematics
§85. Syntactical Sentences in the Literature of the
Special Sciences
§ 86. The Logic of Science is Syntax ....
Bibliography and Index of Authors ....
PAGE
277
281
284
288
292
297
298
302
308
312
31S
322
325
328
331
334
Index of Subjects .
347
PREFACE TO THE ENGLISH EDITION
The present English edition contains some sections which are not
found in the German oiiginal. These are §§ i6fl, 34^-/, 38^-^,
6oa-dy ^\a-e. These twenty-two sections were included in the
manuscript of the German original when it was sent for publication
(in December 1933) but had to be taken out because of lack of
space. The content of § 34^2-2 was, in a slightly different formula-
tion, published in German in the paper Ein Gultigskriterium fur
die Sdtze der klassischen Mathematiky and the content of §§ 60 a-d
and 71 a-d in Die Antinomien und die Unvollstdndigkeit der Mathe-
matik, § 60 of the original has been omitted here, since it w^s only
a shortened substitute for § 60 a-d.
In the Bibliography some less important publications have been
deleted, and others, mainly of the last few years, have been added.
Several smaller additions and corrections have been made. The
more important of these occur at the following points : § 8,
regressive definition; § 12, RI 2 (see footnote); § 14, proofs added
to Theorems 3 and 7 ; § 21, D 29 ; § 22, two insertions in D 64 (see
footnote), D 83 ; § 29, footnote; § 30, PSII 4 (see footnote to § 12) ;
PSII 19, condition added; §48 (see footnote); § 51, definition of
‘ L-consequence ’ ; § 56 (see footnote). Theorems 8 and 9 taken out ;
§ 57, Theorems 2 and 3 corrected, and last paragraph added ;
§62, explanation of ‘Ci2[S2]’*> §§ ^5 definitions of
‘ extensional ’ restricted to closed partial expressions, and Theorem
65.8a added; §67, end of second paragraph. The majority of
these corrections and a number of further ones have been sug-
gested by Dr. A. Tarski, others by J. C. C. McKinsey and W. V.
Quine, to all of whom I am very much indebted for their most
helpful criticisms.
The problem of rendering ihe German terminology was naturally
a most difficult one, in some cases there being no English word in
existence which corresponded exactly to the original, in others the
obvious equivalent being unavailable because of its special associa-
tions in some other system. It was necessary sometimes to appro-
priate for our purposes words which have not previously borne a
technical significance, sometimes to coin entirely new ones. If at
XU PREFACE TO THE ENGLISH EDITION
first sight some of these seem ill at ease or outlandish, I can only
ask the reader to bear in mind the peculiar difficulties involved,
and assure him that no term was chosen without most careful
consideration and the conviction that it would justify itself in use.
To facilitate discussion and reference, the German symbolic
abbreviations have been retained in all the strictly formalized
portions of the book. English equivalents have been substituted
only where they occur in the non-formal text, as mere convenient
abbreviations which are not properly symbolic (e.g. “TN” for
“term-number” instead of the German “GZ”), or as incidental
symbols introduced simply for purposes of illustration (e.g. .“fa”
for “father” instead of the German “ Va”). Wherever a German
abbreviation has been used for the first time, the full German word
has been inserted in brackets ; and in the case of the terms intro-
duced by formal definitions, a complete key to the symbolization
is given in a footnote at the beginning of the respective sections.
I wish to express my best thanks to the Countess von Zeppelin
for the accomplishment of the difficult task of translating this book,
further to Dr. W. V. Quine for valuable suggestions with regard to
terminology, and to Dr. E. C. Graham, Dr. O. Helmer, and Dr. E.
Nagel for their assistance in checking the proofs.
R. C.
FOREWORD
For nearly a century mathematicians and logicians have been
striving hard to make logic an exact science. To a certain extent,
their efforts have been crowned with success, inasmuch as the
science of logistics has taught people how to manipulate with
precision symbols and formulae which are similar in their nature
to those used in mathematics. But a book on logic must contain,
in addition to the formulae, an expository context which, with the
assistance of the words of ordinary language, explains the formulae
and the relations between them ; and this context often leaves much
to be desired in the matter of clarity and exactitude. In recent
years, logicians representing widely different tendencies of thought
have developed more and more the point of view that in this con-
text is contained the essential part of logic ; and that the important
thing is to develop an exact method for the construction of these
sentences about sentences. The purpose of the present work is to
give a systematic exposition of such a method, namely, of the
method of ‘Togical syntax”. (For further details, sec Introduc-
tion, pp. I and 2.)
In our “ Vienna Circle ”, as well as in kindred groups (in Poland,
France, England, U.S.A., and, amongst individuals, even in Ger-
many) the conviction has grown, and is steadily increasing, that
metaphysics can make no claim to possessing a scientific cha-
racter. I'hat part of the work of philosophers which may be held to
be scientific in its nature — excluding the empirical questions which
can be referred to empirical science — consists of logical analysis.
'Phe aim of logical syntax is to provide a system of concepts, a
language, by the help of which the results of logical analysis will
be exactly formidable. Philosophy is to be replaced by the logic of
science — that is to say, by the logical analysis of the concepts and
sentences of the sciences, for the logic of science is nothing other than
the logical syntax of the language of science. That is the conclusion to
which we are led by the considerations in the last chapter of this book.
The book itself makes an attempt to provide, in the form of an
exact syntactical method, the necessary tools for working out the
problems of the logic of science. This is done in the first place by
the formulation of the syntax of two particularly important types
of language which we shall call, respectively, * Language I * and
XIV
FOREWORD
‘ Language 1 1*. Language I is simple in form, and covers a narrow
field of concepts. Language II is richer in modes of expression ; in
it, all the sentences both of classical mathematics and of classical
physics can be formulated. In both languages the investigation
will not be limited to the mathematico-logical part of language —
as is usually the case in logistics — but will be essentially concerned
also with synthetic, empirical sentences. The latter, the so-called
‘ real ’ sentences, constitute the core of science ; the mathematico-
logical sentences are analytic, with no real content, and are merely
formal auxiliaries.
With Language I as an example, it will be shown, in what
follows, how the syntax of a language may be formulated within
that language itself (Part II). The usual fear that thereby con-
tradictions — the so-called ‘epistemological* or ‘linguistic* anti-
nomies — must arise, is not justified.
The treatment of the syntax of Languages I and II will be fol-
lowed by the outline of a general syntax applicable to any language
whatsoever (Part IV) ; and, although the attempt is very far from
attaining the desired goal, yet the task is one of fundamental im-
portance. The range of possible language-forms and, conse-
quently, of the various possible logical systems, is incomparably
greater than the very narrow circle to which earlier investigations
in modern logic have been limited. Up to the present, there has
been only a very, slight deviation, in a few points here and there,
from the form of language developed by Russell which has already
become classical. For instance, certain sentential forms (such as
unlimited existential sentences) and rules of inference (such
as the Law of Excluded Middle), have been eliminated by
certain authors. On the other hand, a number of extensions have
been attempted, and several interesting, many-valued calculi ana-
logous to the two-valued calculus of sentences have been evolved,
and have resulted finally in a logic of probability. Likewise, so-
called intensional sentences have been introduced and, with their
aid a logic of modality developed. The fact that no attempts have
been made to venture still further from the classical forms is per-
haps due to the widely held opinion that any such deviations must
be justified — ^that is, that the new language-form must be proved
to be ‘correct* and to constitute a faithful rendering of ‘the
true logic*.
To eliminate this standpoint, together with the pseudo-problems
FOREWORD
XV
and wearisome controversies which arise as a result of it, is one of
the chief tasks of this book. In it, the view will be maintained that
we have in every respect complete liberty with regard to the forms
of language ; that both the forms of construction for sentences and
the rules of transformation (the latter are usually designated as
“postulates” and “rules vif inference”) may be chosen quite
arbitrarily. Up to now, in constructing a language, the procedure
has usually been, first to assign a meaning to the fundamental
mathematico-logical symbols, and then to consider what sentences
and inferences are seen to be logically correct in accordance with
this meaning. Since the assignment of the meaning is expressed
in words, and is, in consequence, inexact, no conclusion arrived at
in this way can very well be otherwise than inexact and ambiguous.
The connection will only become clear when approached from the
opposite direction : let any postulates and any rules of inference be
chosen arbitrarily; then this choice, whatever it may be, will de-
termine what meaning is to be assigned to the fundamental logical
symbols. By this method, also, the conllict between the divergent
points of view on the problem of the foundations of mathematics
disappears. For language, in its mathematical form, can be con-
structed according to the preferences of any one of the points of view
represented ; so that no question of justification arises at all, but only
the question of the syntactical consequences to which one or other
of the choices leads, including the question of non-contradiction.
The standpoint which we have suggested — we will call it the
Principle of Tolerance (see p. 51) — relates not only to mathe-
matics, but to all questions of logic. From this point of view, the
task of the construction of a general syntax — in other words, of the
definition of those syntactical concepts which are applicable to
languages of any form whatsoever — is a very important one. In
the domain of general syntax, for instance, it is possible to choose
a certain form for the language of science as a whole, as well as for
that of any branch of science, and to state exactly the characteristic
differences between it and the other possible language-forms.
The first attempts to cast the ship of logic off from the terra
firma of the classical forms were certainly bold ones, considered
from the historical point of view. But they were hampered by the
striving after ‘ correctness *. Now, however, that impediment has
been overcome, and before us lies the boundless ocean of un-
limited possibilities.
XVI
FOREWORD
In a number of places in the text, reference is made to the most
important literature on the subject. A complete list has not, how-
ever, been attempted. Further bibliographical information may
easily be obtained from the writings specified. The most import-
ant references are given on the following pages : pp. 96 ff., com-
parison of our Language II with other logical systems; pp. 136 ff.,
□n the symbolism of classes; pp. 158 ff., on syntactical designa-
tions; pp. 253 f., on the logic of modalities; pp. 280 f. and 320 f.
on the logic of science.
For the development of ideas in this book, I owe much to the
stimulation I have received from various writings, letters and con-
versations on logical problems. Mention should here be made ol
the most irhportant names. Above all, I am indebted to the
writings and lectures of Frege. Through him my attention was
drawn to the standard work on logistics — namely, the Principia
Mathematica of Whitehead and Russell. The point of view of the
formal theory of language (known as “ syntax in our terminology)
was first ’developed for mathematics by Hilbert in his “meta-
mathematics”, to which the Polish logicians, especially Ajdukie-
wicz, Lesniewski, Lukasiewicz, and Tarski, have added a “meta-
logic”. For this theory, Godel created his fruitful method of
“ arithmetization ”. On the standpoint and method of syntax, I
have, in particular, derived valuable suggestions from conversa-
tions with Tarski and Godel. I have much for which to thank
Wittgenstein in my reflections concerning the relations between
syntax and the logic of science ; for the divergences in our points
of view, see pp. 282 ff-. (Incidentally, apropos of the remarks made
— e.specially in §17 and §67 — in opposition to Wittgenstein’s
former dogmatic standpoint. Professor SciiMck now informs me
that for some time past, in writings as yet unpublished, Wittgen-
stein has agreed that the rules of language may be chosen with
complete freedom.) Again, I have learned much from the writings
of authors with whom I am not entirely in agreement ; these are,
in the first place, Weyl, Brouwer, and Lewis. Finally, I wish to
express my gratitude to Professor Behmann and Dr. Godel for
having read the manuscript of this book in an earlier draft (1932),
and for having made numerous valuable suggestions towards its
improvement. „ P
INTRODUCTION
§ I. What is Logical Syntax?
By the logical syntax of a language, we mean the formal
theory of the linguistic forms of that language — the systematic
statement of the formal rules which govern it together with the
development of the consequences which follow from these rules.
A theory, a rule, a definition, or the like is to be called formal
when no reference is made in it either to the meaning of the
symbols (for example, the words) or to the sense of the expressions
(e.g. the sentences), but simply and solely to the kinds and order
of the symbols from which the expressions are constructed.
The prevalent opinion is that syntax and logic, in spite of some
points of contact between them, are fundamentally theories of a
very different type. The syntax of a language is supposed to lay
down rules according to which the linguistic structures (e.g. the
sentences) are to be built up from the elements (such as words or
parts of words). The chief task of logic, on the other hand, is sup-
posed to be that of formulating rules according to which judgments
may be inferred from other judgments; in other words, according
to which conclusions may be drawn from premisses.
But the development of logic during the past ten years has shown
clearly that it can only be studied with any degree of accuracy
when it is based, not on judgments (thoughts, or the content of
thoughts) but rather on linguistic expressions, of which sentences
are the most important, because only for them is it possible to lay
down sharply defined rules. And actually, in practice, every
logician since Aristotle, in laying down rules, has dealt mainly
with sentences. But even those modern logicians who agree with
us in our opinion that logic is concerned with sentences, are yet for
the most part convinced that logic is equally concerned, with the
relations of meaning between sentences. They consider that, in
contrast with the rules of syntax, the rules of logic are non-formal.
In the following pages, in opposition to this standpoint, the view
that logic, too, is concerned with the formal treatment of sen-
tences will be presented and developed. We shall see that the
SL a
2
INTRODUCTION
logical characteristics of sentences (for instance, whether a sentence
is analytic, synthetic, or contradictory; whether it is an existential
sentence or not ; and so on) and the logical relations between them
(for instance, whether two sentences contradict one another or are
compatible with one another; whether one is logically deducible
from the other or not ; and so on) are solely dependent upon the
syntactical structure of the sentences. In this way, logic will be-
come a part of syntax, provided that the latter is conceived in a
sufficiently wide sense and formulated with exactitude. The dif-
ference between syntactical rules in the narrower sense and the
logical rules of deduction is only the difference htVNttn formation
rules and transformation rules^ both of which are completely
formulable in syntactical terms. Thus we are justified in desig-
nating as ‘ logical syntax ’ the system which comprises the rules of
formation and transformation.
In consequence of the unsystematic and logically imperfect
structure of' the natural word-languages (such as German or
Latin), the statement of their formal rules of formation and trans-
formation would be so complicated that it would hardly be
feasible in practice. And the same difficulty would arise in the
case of the artificial word-languages (such as Esperanto) ; for, even
though they avoid certain logical imperfections which characterize
the natural word-languages, they must, of necessity, be still very
complicated from the logical point of view owing to the fact that
they are conversational languages, and hence still dependent upon
the natural languages.
For the moment- we will leave aside the question of the formal
deficiencies of the word -languages, and, by the consideration of
examples, proceed to convince ourselves that rules of formation
and transformation are of like nature, and that both permit of being
formally apprehended. For instance, given an appropriate rule, it
can be proved that the word-series “ Pirots karulize elatically ** is a
sentence, provided only that “ Pirots ’’ is known to be a substantive
(in the plural), “karulize” a verb (in the third person plural), and
“ elatically ” an adverb ; all of which, of course, in a well-constructed
language — as, for example, in Esperanto — could be gathered from
the form of the words alone. The meaning of the words is quite
inessential to the purpose, and need not be known. Further, given
an appropriate. rule, the sentence “A karulizes elatically” can be
§1. WHAT IS LOGICAL SYNTAX? 3
deduced from the original sentence and the sentence “A is a
Pirot” — again provided that the type to which the individual
words belong is known. Here also, neither the meaning of the
words nor the sense of the three sentences need be known.
Owing to the deficiencies of the word-languages, the logical
syntax of a language of this kind will not be developed, but, in-
stead, we shall consider the syntax of two artificially constructed
symbolic languages (that is to say, such languages as employ
formal symbols instead of words). As a matter of fact, throughout
all modern logical investigations, this is the method used; for only
in a symbolic language has it proved possible to achieve exact
formulation and rigid proofs. And only in relation to a constructed
symbolic language of this kind will it be possible to lay down a
system of rules at once simple and rigid — which alone will enable
us to show clearly the characteristics and range of applicability
of logical syntax.
The sentences, definitions, and rules of the syntax .of a language
are concerned with the forms of that language. But, now, how
are these sentences, definitions, and rules themselves to be cor-
rectly expressed? Is a kind of super-language necessary for the
purpose? And, again, a third language to explain the syntax of this
super-language, and so on to infinity? Or is it possible to formulate
the syntax of a lang\iage within that language itself? The obvious
fear will arise that in the latter case, owing to certain reflexive
definitions, contradictions of a nature seemingly similar to those
which are familiar both in Cantor’s theory of transfinite aggregates
and in the pre-Russellian logic might make their appearance.
But we shall see later that without any danger of contradictions or
antinomies emerging it is possible to express the syntax of a lan-
guage in that language itself, to an extent which is conditioned by
the wealth of means of expression of the language in question.
However, we shall not at first concern ourselves with this pro-
blem, important though it is. We shall proceed, instead, to construct
syntactical concepts relating to the languages we have chosen, and
postpone, for a while, the question as to whether we are able oi not
to express the rules and sentences based on these concepts in that
language itself. In the first stages of a theory, such a naive approach
seems always to have proved the most fruitful. For instance, geo-
metry, arithmetic, and the differential calculus all appeared first.
4
INTRODUCTION
and only much later (in some cases, hundreds of years after) did
epistemological and logical discussions of the already developed
theories ensue. Hence we shall start by constructing the syntax,
and then, iater on, proceed to formalize its concepts and thereby
determine its logical character.
In following this procedure, we are concerned with two lan-
guages : in the first place with the language which is the object of
our investigation — ^we shall call this the object-language — and,
secondly, with the language in which we speak about the syntactical
forms of the object-language — we shall call this the syntax-
language. As we have said, we shall take as our object-languages
certain symbolic languages; as our syntax-language we shall at
first simply use the English language with the help of some
additional Gothic symbols.
§ 2. Languages as Calculi
By a calculus is understood a system of conventions or rules of
the following kind. These rules are concerned with elements — the
so-called symbols — about the nature and relations of which
nothing more is assumed than that they are distributed in various
classes. Any finite series of these symbols is called an expression
of the calculus in question.
The rules o/ the calculus determine, in the first place, the con-
ditions under which an expression can be said to belong to a cer-
tain category of expressions; and, in the second place, under what
conditions the transformation of one or more expressions into
another or others may be allowed. Thus the system of a language,
when only the formal structure in the sense described above is
considered, is a calculus. The two different kinds of rules are those
which we have previously called the rules of formation and trans-
formation — namely, the syntactical rules in the narrower sense
(e.g. '*An expression of this language is called a sentence when it
consists, in such and such a way, of symbols of such and such a
kind, occurring in such and such an order”), and the so-called
logical laws of deduction (e.g. ‘*If a sentence is composed of
symbols combined in such and such a way, and if another is
composed of symbols combined in such and such another way,
then the second can be deduced from the first”). Further, every
5
§ 2. LANGUAGES AS CALCULI
well -determined mathematical discipline is a calculus in this sense.
But the system of rules of chess is also a calculus. The chessmen
are the symbols (here, as opposed to those of the word-languages,
they have no meaning), the rules of formation determine the posi-
tion of the chessmen (especially the initial positions in the game),
and the rules of transformation determine the moves which are per-
mitted — that is to say, the permissible transformations of one
position into another.
In the widest sense, logical syntax is the same thing as the con-
struction and manipulation of a calculus; and it is only because
languages are the most important examples of calculi that, as a
rule, only languages are syntactically investigated. In the majority
of calculi (even in those which are not languages in the proper
sense of the word), the elements are written characters. The term
‘symbol* in what follows will have the same meaning as the word
‘ character*. It will not be assumed that such a symbol possesses a
meaning, or that it designates anything.
When we maintain that logical syntax treats language as a cal-
culus, we do not mean by that statement that language is nothing
more than a calculus. We only mean that syntax is concerned with
that part of language which has the attributes of a calculus — that
is, it is limited to the formal aspect of language. In addition, any
particular language has, apart from that aspect, others which may
be investigated by other methods. For instance, its words have
meaning; this is the object of investigation and study for sema-
siology. Then again, the words and expressions of a language have
a close relation to actions and perceptions, and in that connection
they are the objects of psychological study. Again, language con-
stitutes an historically given method of communication, and thus of
mutual influence, within a particular group of human beings, and
as such is the object of sociology. In the widest sense, the science
of language investigates languj’ges from every one of these stand-
points: from the syntactical (in our sense, the formal), from the
semasiological, from the psychological, and from the sociological.
We have already said that syntax is concerned solely with the
formal properties of expressions. We shall now make this assertion
more explicit. Assume that two languages {Sprachen)^ and Sg,
use different symbols, but in such a way that a one-one corre-
spondence may be established between the symbols of Si and those
6
INTRODUCTION
of Sg SO that any syntactical rule about becomes a syntactical
rule about Sg if, instead of relating it to the symbols of Sj , we re-
late it to the correlative symbols of Sg ; and conversely. Then,
although the two languages are not alike, they have the same
formal structure (we call them isomorphic languages), and syntax
is concerned solely with the structure of languages in this sense.
From the syntactical point of view it is irrelevant whether one of
two symbolical languages makes use, let us say, of the sign *&’,
where the other uses ‘ * (in word-languages : whether the one uses
‘ and * and the other ‘ und *) so long as the rules of formation and
transformation are analogous. For instance, it depends entirely on
the formal structure of the language and of the sentences involved,
whether a certain sentence is analytic or not ; or whether one sen-
tence is deduciblc from another or not. In such cases the design
(visual form, Gestalt) of the individual symbols is a matter of in-
difference. In an exact syntactical definition, no allusion will be
made to this design. Further, it is equally unimportant from the
syntactical point of view, that, for instance, the symbol ‘and'
should be specifically a thing consisting of printers' ink. If we
agreed always to place a match upon the paper instead of that
particular symbol, the formal structure of the language would
remain unchanged.
It should now be clear that any series of any things will equally
well serve as terms or expressions in a calculus, or, more parti-
cularly, in a language. It is only necessary to distribute the things
in question in particular classes, and we can then construct ex-
pressions having the form of series of things, put together according
to the rules of formation. In the ordinary languages, a series of
symbols (an expression) is either a temporal series of sounds, or a
spatial series of material bodies produced on paper. An example of
a language which uses movable things for its symbols is a card-
index system ; the cards serve as the object-names for the books of
a library, and the riders as predicates designating properties (for
instance, ‘lent', ‘at the book-binders', and such like); a card with
a rider makes a sentence.
The syntax of a language, or of any other calculus, is concerned,
in general, with the structures of possible serial orders (of a definite
kind) of any elements whatsoever. We shall now distinguish
between pure and descriptive syntax. Pure syntax is concerned
§2. LANGUAGES AS CALCULI 7
with the possible arrangements, without reference either to
the nature of the things which constitute the various elements,
or to the question as to which of the possible arrangements
of these elements are anywhere actually realized (that is to say,
with the possible forms of sentences, without regard either to the
designs of the words of which the sentences are composed, or to
whether any of the sentences exist on paper somewhere in the
world). In pure syntax only definitions are formulated and the
consequences of such definitions developed. Pure syntax is thus
wholly analytic, and is nothing more than combinatorial analysis yOVy
in other words, the geometry of finite, discrete, serial structures of
a particular kind. Descriptive syntax is related to pure syntax as
physical geometry to pure mathematical geometry ; it is concerned
with the syntactical properties and relations of empirically given
expressions (for example, with the sentences of a particular book).
For this purpose — just as in the application of geometry — it is
necessary to introduce so-called correlative definitions, by means
of which the kinds of objects corresponding to the different kinds
of syntactical elements are determined (for instance, “material
bodies consisting of printers’ ink of the form ‘ V ' shall serve as dis-
junction symbols”). Sentences of descriptive syntax may, for in-
stance, state that the fourth and the seventh sentences of a parti-
cular treatise contradict one another; or that the second sentence
in a treatise is not syntactically correct.
When we say that pure syntax is concerned with the forms of
sentences, this ‘concerned with’ is intended in the figurative
sense. An analytic sentence is not actually “concerned with”
anything, in the way that an empirical sentence is ; for the analytic
sentence is without content. The figurative ‘concerned with’ is
intended here in the same sense in which arithmetic is said to be
concerned with numbers, or pure geometry to be concerned with
geometrical constructions.
We see, therefore, that whenever we investigate or judge a
particular scientific theory from the logical standpoint,, the results
of this logical analysis must be formulated as syntactical sentences y
either of pure or of descriptive syntax. The logic of science (logical
methodology) is nothing else than the syntax of the language of
science. This fact will be shown clearly in the concluding chapter
of 'his book. The syntactical problems acquire a greater significance
8
INTRODUCTION
by virtue of the anti-metaphysical attitude represented by the
Vienna Circle. According to this view, the sentences of meta-
physics are pseudo-sentences which on logical analysis are proved
to be either empty phrases or phrases which violate the rules of
syntax. Of the so-called philosophical problems, the only ques-
tions which have any meaning are those of the logic of science. To
share this view is to substitute logical syntax for philosophy. The
above-mentioned anti-metaphysical attitude will not, however,
appear in this book either as an assumption or as a thesis. The in-
quiries which follow are of a formal nature and do not depend in
any way upon what is usually known as philosophical doctrine.
The method of syntax which will be developed in the following
pages will not only prove useful in the logical analysis of scientific
theories — it will also help in the logical analysis of the word-
languages. Although here, for the reasons indicated above, we shall
be dealing with symbolic languages, the syntactical concepts and
rules — not in detail but in their general character — may also be
applied to the analysis of the incredibly complicated word-
languages. The direct analysis of these, which has been
prevalent hitherto, must inevitably fail, just as a physicist
would be frustrated were he from the outset to attempt to relate
his laws to natural things — trees, stones, and so on. In the first
place, the physicist relates his laws to the simplest of constructed
forms; to a thin straight lever, to a simple pendulum, to puncti-
form masses, etc. Then, with the help of the laws relating to
these constructed forms, he is later in a position to analyze into
suitable elements the complicated behaviour of real bodies, and
thus to control them. One more comparison : the complicated con-
figurations of mountain chains, rivers, frontiers, and the like are
most easily represented and investigated by the help of geographical
co-ordinates — or, in other words, by constructed lines not given in
nature. In the same way, the syntactical property of a particular
word-language, such as English, or of particular classes of word-
languages, or of a particular sub-language of a word-language, is
best represented and investigated by comparison with a constructed
language which serves as a system of reference. Such a task, how-
ever, lies beyond the scope of this book.
§ 2. LANGUAGES AS CALCULI
9
Terminological Remarks
The reason for the choice of the term * (logical) syntax * is given
in the introduction. The adjective ‘logical’ can be left out where
there is no danger of confusion with linguistic syntax (which is not
pure in its method, and does not succeed in laying down an exact
system of rules), for example, in the text of this book and in logical
treatises in general.
As the word itself suggests, the earliest calculi in the sense
described above were developed in mathematics. Hilbert was the
first to treat mathematics as a calculus in the strict sense — i.e. to
lay down a system of rules having mathematical formulae for
their objects. This theory he called metamathematics ^ and his
original object in developing it was to attain the proof of the free-
dom from contradiction of classical mathematics. Metamathe-
matics is — when considered in the widest sense and not only from
the standpoint of the task just mentioned — the syntax of the mathe-
matical language. In analogy to the Hilbertian designation, the
Warsaw logicians (Lukasiewicz and others) have spoken of the
‘ meta-propositional calculus’, of metalogicy and so on. Perhaps
the word ‘ metalogic ’ is a suitable designation for the sub-domain
of syntax which deals with logical sentences in the narrower sense
(that is, excluding the mathematical ones).
The term semantics is used by Chwistek to designate a theory
which he has constructed with the same object as our syntax, but
which makes use of an entirely different method (of this we shall
say more later). But since, in the science of language, this word
is usually taken as synonymous with ‘ semasiology ’ (or ‘ theory of
meaning’) it is perhaps not altogether desirable to transfer it to
syntax — that is, to a formal theory which takes no account of
meanings. (Compare: Br^al, Essai de s^mantique. Science des
significations. Paris, 1897, 5th edn. 1921, p. 8: “La science, que
j’ai propose d’appeler la S^mantique”, with footnote: ErfyLavriK^
la science des significations’’.)
The designation sematology may (following Biihler) be retained
for the empirical (psychological, sociological) theory of the appli-
cation of symbols in the widest sense. The empirical science of
language is thus a sub-domain of sematology. But it must be
distinguished from semasiology which, as a part of the science of
language, investigates the m^^ning of the expressions of the
historically given languages.
PART !
THE DEFINITE LANGUAGE I
A. RULES OF FORMATION FOR LANGUAGE I
§ 3. Predicates and Functors
The syntactical method will here be developed in connection
with two particular symbolic languages taken as object-languages.
The first of these languages — we shall call it Language I, or,
briefly, I — includes, on the mathematical side, the elementary
arithmetic of the natural numbers to a certain limited extent,
roughly corresponding to those theories which are designated as
constructivist, finitist, or intuitionist. The limitation consists pri-
marily in the fact that only definite number-properties occur —
that is to say, those of which the possession or non-possession by
any number whatsoever can be determined in a finite number of
steps according to a fixed method. It is on account of this limita-
tion that we call I a definite language, although it is not a definite
language in the narrower sense of containing only definite, that is
to say, resoluble (i.e. either demonstrable or refutable) sentences.
Later on, we shall be dealing With Language II, which includes
Language I within itself as a sub-language. Language II contains
in addition indefinite concepts, and embraces both the arithmetic
of the real numbers and mathematical analysis to the extent to
which it is developed in classical mathematics, and further the
theory of aggregates. Languages I and II do not only include
mathematics, however; above all, they afford the possibility of
constructing empirical sentences concerning any domain of objects.
In II, for instance, both classical and relativistic physics can be
formulated. We attach special importance to the syntactical treat-
ment of the synthetic (not purely logico -mathematical) sentences,
which are usually ignored in modern logic. The mathematical
sentences, considered from the point of view of language as a
whole, are only aids to operation with empirical, that is to say,
non-mathematical, sentences.
12
PART I. THE DEFINITE LANGUAGE I
In Part I the syntax of Language I will be formulated. Here,
the English language, supplemented by a few Gothic symbols, will
be used as the syntax-language. In Part II the syntax of Lan-
guage I will be formalized, that is, it will be expressed in the form
of a calculus-language ; and this will be done in Language I itself.
In Part III the syntax of the richer Language II will be developed,
but only by the simpler method of a word-language. In Part IV
we shall abandon the object-languages I and II, and create a general
syntax which will be applicable to all languages of every kind.
For the understanding of the following chapters, a previous
knowledge of the elements of logistics (symbolic logic) is desirable,
although not absolutely necessary. Further details supplementing
the short -explanations given here are to be found in the regular
expositions of the sentential calculus and the so-called functional
calculus. See: Hilbert [Logtk]; Carnap [Logistik]; Lewis [Logic],
A language which is concerned with the objects of any domain
may designate these objects either hy proper names or by systematic
positional co-ordinates y that is by S3rmbols which show the place of
the objects in the system, and, thereby, their positions in relation
to one another. Examples of positional symbols are, for instance,
house-numbers, in contradistinction to the individual names (such
as ‘The Red Lion*) which were customary in earlier days;
Ostwald’s designation of colours by means of letters and figures, as
opposed to tbeir differentiation by means of colour-names (‘ blue*,
etc.) ; the designation of geographical places by their latitude and
longitude, instead of by proper names (‘Vienna*, ‘ Cape of Good
Hope *) ; and the customary designation of space-time points by
four co-ordinates (space and time co-ordinates — four real num-
bers) in physics. The method of designation by proper names is the
primitive one ; that of positional designation corresponds to a more
advanced stage of science, and has considerable methodological
advantages over the former. We shall call a language (or sub-
language) which denotes the objects belonging to the domain with
which it is concerned by positional designations, a co-ordinate-
language y in contradistinction to the name-languages.
Up to now it has been usual in symbolic logic to use name-
languages, the objects being, for the most part, designated by the
names ‘a’, ‘L*, etc. (corresponding to the designations ‘moon*,
‘Vienna*, ‘Napoleon*, of the word-languages). Here, we shall
13
§3. PREDICATES AND FUNCTORS
take co-ordinate-languages for our object-languages, and, speci-
fically, in Language I, we shall use the natural numbers as co-
ordinates. Let us consider, as a domain of positions, a one-dimen-
sional series with a definite direction. If ‘a’ designates a position
in this series, then the next position will be designated by ‘a* *. If
the initial position is designated by ‘O’, then the succeeding posi-
tions will obviously be designated by ‘O* ’, ‘O" ’, and so on. We call
such expressions accented expressions. Since, however, for the
representation of higher positions, they entail a certain amount of
inconvenience, we shall, for the purpose of abbreviation, introduce
the usual number-symbols by definition. Thus : ‘ 1 ’ for ‘ O' ‘ 2 ’ for
‘0“*, and so on. If we wish to indicate the positions in a two-,
three- , or 12-dimensional domain, we use ordered couples or triads
or Ti-ads of number-symbols.
In order to express a property of an object, or of a position, or a
relation between several objects or positions, predicates are used.
Examples: (i) Let ‘Blue (3)’ have the meaning: “the position 3 is
blue”; in a name-language: ‘Blue (a)’ is “the object a' is blue”.
(2) Let ‘ Wr (3,5) * mean : “the position 3 is warmer than the posi-
tion 5”; in a name-language ‘Wr(a,b)’: “the body a is warmer
than the body b ” ; ‘Fa (a,b) ’ : “ the person a is father of the person
b”, and so on. (3) Let ‘T( 0 , 8,4,3)’ mean: “the temperature at
the position 0 is as much higher than at the position 8 as the tem-
perature at the position 4 is higher than at the position 3 ”. In the
above examples, ‘blue* is a one-termed predicate; ‘Wr* a two-
termed predicate ; ‘ T ’ a four-termed predicate. In ‘ Wr (3 , 5) *, ‘ 3 ’
is called the first, and ‘ 5 * the second argument of ‘ Wr*. We dis-
tinguish two classes of predicates : the predicates in the examples
cited above express (as we usually say) empirical properties or
relations. We call these descriptive predicates ^ and distinguish them
from logical predicates ^ which are those which (as we usually say)
express logico-mathematical properties or relations. The following
are examples of logical predicates : ‘ Prim (5) * means : “ 5 is a prime
number ’* ; ‘ Gr (7, 5) * : “ 7 is greater than 5 ”, or : “ the position 7 is
a higher position than the position 5 The exact definition of the
syntactical concepts ‘descriptive* and ‘logical* will be given
later, without reference to meaning as in the present inexact
explanation. [The designation ‘predicate*, which was formerly
applied ojily in cases involving one term, will here, following the
14 PART I. THE DEFINITE LANGUAGE I
example of Hilbert, be applied also in cases involving more than
one term; the use of a common word to cover both cases has
proved itself to be far more practical.]
Predicates are, so to speak, proper names for the properties of
positions. We have designated positions by means of systematic
order-symbols — namely, number-symbols. In like manner we may
also designate their properties by number-symbols. Instead of
colour-names, colour-numbers (or triads of such numbers) may be
used; instead of the inexact designations ‘warm*, ‘cool*, ‘cold*,
and so forth, we can now use temperature-numbers. This has not
only the advantage that much more exact information can be
given, but, in addition, a further advantage which is of decisive
importance for science — namely, that only by means of this
“arithmetization’* is the formulation of universal laws (for ex-
ample: that of the relation between temperature and expansion,
or between temperature and pressure) rendered possible. In order
to express properties or relations of position by means of numbers,
we shall use functors. For instance: let ‘te* be the temperature
functor ; * te (3) = s * then means : “ the temperature at the position
3 is 5**; if we take the functor ‘tdiff* to represent temperature
difference, then ‘ tdiff (3,4) = 2* means: “the difference of the
temperatures at positions 3 and 4 equals 2**. Besides such descrip-
tive functors j we make use also of logical functors. For example :
‘ sum (3,4)’ has the meaning : “ 3 plus 4 ** ; ‘ fak (3) * is equivalent to
“3!”. ‘sum* is a two-termed logical functor, ‘fak’ {Fakultdt) a
one-termed logical functor. Here also in the expression ‘ sum (3,4)*,
‘3’ and ‘4’ are called arguments', in‘te(3) = 5*, ‘3’ is called the
argument for ‘te’, and ‘5* is called the value of ‘te* for the
argument ‘ 3 *.
An expression which in any way designates a number (deter-
mined or undetermined), we call a numerical expression (exact
definition on p. 26). Examples are: ‘O’, ‘ 0 »*, ‘3*, ‘te(3)’,
‘sum (3,4)’. An expression which corresponds to a propositional
sentence of a word-language we call a sentence (definition on p. 26).
Examples arc: ‘Blue (3)*, ‘Prim (4)’. An expression is called de-
scriptive (definition on p. 25) when either a descriptive predicate
or a descriptive functor occurs in it ; otherwise it is called logical
(definition on p. 25).
§ 4- SYNTACTICAL GOTHIC SYMBOLS
IS
§ 4. Syntactical Gothic Symbols
The two symbols ‘ a ’ and ‘ a ’ occur at different places on this
page. They are therefore different symbols (not the same symbol);
but they are equal (not unequal). The syntactical rules of a language
must not only determine what things are to be used as symbols,
but also under what circumstances these symbols are to be re-
garded as syntactically equal. Very often, symbols which are un-
like in appearance are stated to be syntactically equal : for example,
in ordinary language, *z’ and ‘5’. [Such a declaration of equality
does not always necessarily mean that the two symbols ate to be
used indiscriminately. There may be differences in usage which
depend on non-syntactical factors. For instance, it is customary
not to use ‘ z * and ‘ 3 * in the same context : one writes nearly always,
either ‘zebra* or ‘3Cbra*, not ‘gebra*.] As they are used in this
book, ‘ z * and ‘ 3 * are syntactically unequal. On the other hand, we
shall regard ‘(*, ‘ (*, ‘[’, ‘ [* as equal symbols, and likewise the
corresponding closing brackets. The differentiation of small and
large, round and square brackets in the expressions of our object-
language is therefore syntactically irrelevant. Such a differentiation
is introduced solely for the convenience of the reader. Further, in
our system (in contrast with RusselFs) the symbols ‘ = * and ‘ = * are
held to be equal. We could write ‘ = ’ throughout, but, again tor the
convenience of the reader, when ‘ = * occurs between sentences (and
not between numerical expressions) we usually write ‘ = * instead.
We shall call two expressions equal expressions when their
corresponding symbols are equal symbols. If two symbols, or two
expressions, are equal (syntactically), then we say also that they
have the same syntactical design. But that does not in any way
prevent their having different visual shapes, as, for example, in the
case of ‘ ( * and ‘ [ *, or ‘ = ’ and ‘ ; or differing in colour, or any
other characteristics that are syntactically irrelevant.
Nearly all the investigations carried out in this book are con-
cerned with pure (not descriptive) syntax; and thus have to do,
not with expressions as spatially separate things, but only with
their syntactical equality or inequality, and hence with their
syntactical design. Whatever is stated of any one expression applies
at the same time to every other equal expression, and may, ac-
i6
PART I. THE DEFINITE LANGUAGE I
cordingly, be predicated of the expressional design. Therefore, for
the sake of brevity, we shall often speak simply of ‘ expression * or
‘symbor, instead of ‘expressional design' or ‘symbolic design'.
[For instance, instead of saying: “ in the expression ‘ Q (3, 5) * (and
hence in every equal expression) a symbol like the symbol ‘3'
occurs", we say more briefly: “in every expression of the design
* Q (3, 5) * a symbol of the design ‘ 3 * occurs " ; or, still more simply,
“in the expression ‘Q(3,s)’, the symbol ‘3' occurs".] In the
domain of pure syntax, this simplified form of speech cannot lead
to ambiguity.
Symbols of the five kinds enumerated below occur in I. (Ex-
planations follow later.)
I . Eleven individual symbols (symbol-designs) :
«/i iy i , i,» « . iy, *1 « ^ I
9 < -1 ’ <
> d y
K'.
The following four categories, to each of which an unlimited
number of symbols may belong:
2. Th <5 (numerical) variables the definitions
of §§20-24, also ‘/e’, ‘ ‘f’).
3. The constant numerals (e.g. ‘O’, ‘ i ', ‘2', etc.); the symbols
belonging to groups (2) and (3) are called numerals.
4. The predicates (groups of letters with initial capitals, e.g.
‘Prim*, also‘P',‘Q', ‘R’).
5. The functors (groups of letters with small initial letters, e.g.
‘sum’).
A symbol which is not a variable is called a constant. An ex-
pression of I consists of an ordered series of symbols of I , of which
the number is finite (but which may also be either 0 or i ; that is
to say, an expression may either be empty or consist of one symbol
only).
By a syntactical form (or, shortly: a form) we understand any
kind or category of expressions which is syntactically determined
(that is to say, determined only with reference to the serial order
and the syntactical category of the symbols ; and not by any non-
syntactical conditions such as place, colour, etc.). The form of a
certain expression can be specified more or less exactly : the most
accurate specification is that which gives the design of the ex-
pression; the most inaccurate, that which merely states that it is
an expression.
17
§4- SYNTACTICAL GOTHIC SYMBOLS
We shall introduce an abbreviated method for writing down
statements about form. For instance, in the language of words, we
cairmake the following statement about the form of the expression
‘Prim(jic)*. ‘'This expression consists of a predicate, an opening
bracket, a variable, and a closing bracket, written in this order.”
Instead of this we shall write more briefly : “ this expression has the
form pr (3) **• This method of the use of the Gothic symbols consists
in introducing syntactical names to represent symbol-categories;
the syntactical description of form is then effected simply by placing
these syntactical names one after the other. We shall designate the
symbols (of all designs) by ‘q*; the variables (numerical) by
‘ 3 ’ (Zahlvariable) ; the symbol (symbol-design) ‘ 0 * by ‘ nu * (null ) ;
the numerals in general by ‘ 33 * (Zahlzeichen) ; the predicates by ‘ pt *
(and, specifically, the one-termed, two-termed, n-termed pjredi-
cates by ‘ pr^ *, ‘ pt^ *, ‘ pr” *, respectively) ; the functors by ‘ fu * (and,
specifically, etc.). For the syntactical designation of the
eleven individual symbols, we shall use the symbols 'themselves,
and in addition for the two-termed junction’-symhols (Verkniip-
fungszeicherC) — ‘V’, ‘ d ‘ = * — the designation ‘pcrfn\ Thus,
for instance, in ‘ Prim (x) ’, * ( ' is a symbol of the object-language ;
on the other hand, in *pr(3)*, *(’ is a symbol of the syntax-
language which serves as a syntactical name for that symbol in the
object-language, and is, accordingly, nothing else than an abbre-
viation for the English words ‘opening bracket \ When a symbol
is used in this way as a name for itself (or, more precisely, as a
name for its own symbol-design), we call it an autonymous symbol
(see §42). No ambiguity can arise as a result of the double use
of the symbols ‘ ( ’, etc., since these symbols only occur autony-
mously in connection with Gothic letters. Whenever we wish to
differentiate different symbols of the same kind by their syntactical
designations we use indices. For instance: ‘ P (jc,_y , ac) * has the
form pr (3,3,3), or, more exactly, the form pr® (31,32, 3 i)*
most important kinds of expressions we shall also use syntactical
symbols (with capital letters). Expressions (of any form) we
designate by ‘ 91 * (Ausdruck), numerical expressions by ‘ 3 *
(Zahlausdruck)y sentences by ‘ S * (Satz). Other designations will
be introduced later. Here, also, we make use of indices in order to
indicate the equality of expressions. In a sentence of the form
(< 3 vS)dS, the three constituent sentences may be equal or
SL
3
l8 PART I. THE DEFINITE LANGUAGE I
unequal; in a sentence of the form (SiVSg)^®!, the first and
third of the constituent sentences are equal sentences.
By means of the indices ‘b ’ and *1’, it is possible to indicate that
a symbol is descriptive or logical respectively. For instance, ‘fui*
designates the logical funaors, ‘ 3b * the descriptive numerical ex-
pressions. Instead of writing “a symbol (or expression) of the
form...§ **, we often write for short: ‘a...*; for example, instead
of “a two-termed logical functor”, we write briefly: ‘an fU|**;
similarly ‘a 3*» ‘an 9lb*, and so on.
In what follows the Gothic symbols will be used in connection
with the English text ; in the later construction of the syntax of I,
which is not given in a word-language but by means of further
symbols, these symbols do not occur.
The chief object of the method of Gothic symbols is to protect
us from the incorrect mode of expression, very frequent in both
logical and mathematical writings, which makes no distinction be-
tween symbols and that which is symbolized. For instance, we
find “ in this or in that place, x—y occurs **, where the correct form
would be “ . . .‘x=y ’ occurs**, or .31 = 32 occurs**. If an ex-
pression of the object-language is being discussed, then either this
expression must be written in inverted commas, or its syntactical
designation (without inverted commas) must be used. But if the
syntactical designation is what we are talking about, then it, in
turn, must be put into inverted commas. Later on we shall show
how very easily the neglect of this rule, and the failure to dif-
ferentiate between symbols and the objects designated by them,
leads to error and obscurity (§§ 41, 42).
§ 5. The Junction Symbols
The one-termed or two-termed junction symbols are used to
construct a new sentence out of one or two sentences respectively.
In a strictly formally constructed system, the meaning of these
symbols — as will be discussed more fully later — arises out of the
rules of transformation. In order to facilitate the understanding of
them, we shall provisionally explain their meaning (and similarly
that of other symbols) by less exact methods ; first, by an approxi-
mate translation into words of the English language, and secondly,
with more precision, by means of the so-called truth-value tables.
§ S. THE JUNCTION SYMBOLS 1 9
called the negation of (S^); (Si)V(S2), (Si)«(S2)»
(Si) D (S2), (Si) = (S2) are called^ respectively, the disjunction,
conjunction, implication, equivalence (or equation) of Si and S2 »
in which Si and S2 are called terms.
The translations of these symbols are as follows : ‘ not ’ ; ‘or * (in
the non-exclusive sense); ‘and*; ‘not. . .or. . . ’ (sometimes also
translatable by ‘if. . .then. . . *); ‘either. . .and. . ., or not. . .and
not . . . *. We shall usually write the symbol design ‘ = * in the form
‘ = * where it occurs between sentences (not between numerical ex-
pressions) ; ‘ = ’ and ‘ = * count, therefore, as equal symbols, i.e. as
symbols of the same syntactical design.
In the majority of accepted systems, a special symbol of equiva-
lence is used, in addition to the symbol of identity or equality ‘ = *.
(For instance, Russell uses ‘ = *, Hilbert ‘ — *.) We, on the other
hand, both in Language I and in Language II use only onfe sym-
bolic design (but for the easier comprehension of the reader, we use
two kinds of figures). As we shall see later (pp. 244 f.), this method
is both admissible and useful for extensional languages such as
I and 11 .
In what follows, for the sake of brevity in writing down any
symbolic expressions that occur either in the object- or in the
syntax- languages, we shall (as is customary) leave out the brackets
surrounding a partial expression (which may be either a sentence
or the syntactical designation of a sentence) in the following cases :
1 . When 9Ii consists of one letter only.
2. In the relation ^(2li), or Dcrfn(?Ii), or (?li)pcrfn, when %
begins either with ‘ <^ *, or with a pr, or with an operator
(see below).
3. When is a disjunctive term, and is itself a disjunction.
4. When 3Ii is a conjunctive term, and is itself a conjunction.
5. When is an operand and itself begins with an operator
(of this more later).
Thus instead of ‘ (^(Si))v(S2)* [but not instead of
‘~((S,)v(S,))’], we write for short: ‘^SiVS2*; similarly:
‘S1VS2VS3*, ‘Si.S2-S3’.
This simplification will, however, be used here only for the
practical purpose of writing down the expressions — the formula-
tion of syntactical definitions and rules will be referred to the ex-
pressions with no brackets omitted.
There are, obviously, four possibilities in connection with the
20
PART I. THE DEFINITE LANGUAGE I
truth and falsehood of two sentences, Si and S2 • These will be
represented by the four lines of the truth-value table given below.
The table shows in which of these four cases the junction sentence
is true and in which it is false ; for instance, the disjunction is false
only in the fourth case, otherwise it is true.
®,s,
SiVSj
Si.S,
SiOS,
(I)
T T
T
T
T
T
(2)
T F
T
F
F
F
(3)
F T
T
F
T
F
( 4 )
F F
F
F
T
T
The two-line table below is the table for negation.
S,
-61
(1)
T
F
(2)
F
T
With the help of the above tables, the truth value of a multiple
compound sentence can easily be ascertained for the different
cases by first of all determining the values for the component
partial-sentences, and then proceeding step by step to the whole
sentence. Thus, for instance, it can be determined that for
V Sgi the same truth distribution T, F, T, T holds good as
for the case of implication ; and from this we get the translation
into words: ‘not. . .or. . . for implication. Further it can also be
established that Si D (Si-V S2) has the truth-value distribution,
T, T, T, T, and is thus unconditionally true whether Si and S2
be true or false. Later on we shaU call such sentences analytic
sentences,
§ 6. Universal and Existential Sentences
Here we shall again give the meanings of expressions by means
first of translation and then of a statement of the truth-conditions.
For instance, let ‘ Red ’ be a ptb ; ‘ Red (3) * will then mean : “ The
position 3 is red.” Now, let * (jk?) (Red (jc))* mean : “ Every position
is red”; and ‘Q x) (Red {x)y: “At red”, and
therefore: “There is (at least) on^ position uiit Besides
these ordinary forms of sentenc^ we shall introduce thel^llowing.
§6. UNIVERSAL AND EXISTENTIAL SENTENCES 21
(x)3 (Red(»))’ will mean the same as: ‘ Red ( 0 ). Red (i). Red (2).
Red (3)*, that is: “Every position up to 3 is red”; ‘(3 a:)3
(Red{x)y will mean the same as ‘Red(0)VRed(i)VRed(2)V
Red (3)*, or “There is one position up to 3 which is red.“
The expressions which occur at the beginning of the sentences
above, namely : ‘ (x) ‘ (3 jc)**, ‘ (x) 3 ‘ (3 3 are called the un-
limited universal operator, the unlimited existential operator,
the limited universal operator, and the limited existential operator
respectively. In the two limited operators, ‘3’ is called the
limiting expression of the operator, and in all four of the operators
X * is called the operator-variable. * Red (x) * is called the operand
(belonging to the operator). In Language I, only limited operators
occur ; we shall not make use of the unlimited operators till later,
in Language II.
If 9 Ii and 3I2 are operators, then, instead of writing ( 9 l 2 (®)),
we shall write simply ?Ii?l2(S). (Compare p. 19, condition 5.)
A variable (or the symbol-design of a variable), 31, is called
bound at a certain position in (whether a symbol of the design
3i occurs at this position or not) when there is a (proper or im-
proper) partial-sentence of which contains this position and
has the form ^ (S), where 312 operator having the operator-
variable 3i.
A variable 32 which occurs at a certain position in 3 Ii is called
free at this position in 3Ii when 32 is not bound at this position
in 3 Ii. Example: Let Si have the form : S2V S3V S4 ; and specifi-
cally the design : ‘ Pj (jc) V (oc) 5 (P2 ( ^,j) ) V P3 (a:) \ At all positions
of S3, ‘ a: ’ is bound in S3 and therefore also in Si ; in Si the first
and the fourth * x' and the ‘jy’ are free. If a variable which is
free in 3 ti occurs in 3Ii, then 3li is called open; otherwise it is
called closed.
In order to express unlimited universality^ fret variables will be
used in I. For example, let Ss be ‘sum(A:,j') = sum(y,A:)\ This
will mean: “For any two numbers, the sum of the first and the
second is always equal to the sum of the second and first.*' If S5
is true, then so is every sentence arising out of Ss as the result
of substituting any arbitrary numerical expressions for ‘a;* and
for instance, ‘sum (3, 7) = sum (7, 3)* (SJ. [Thus, in our
system, the so-called sentential functions also are ranked as sen-
tences. Our classification into closed and open sentences corre-
22
PART I. THE DEFINITE LANGUAGE I
spends to the usual classification into sentences and sentential
functions.]
In the use of free variables for expressing unlimited universality,
our language agrees with that of Russell. But when Russell, in his
explanatory text [Princ. Math. I] says that a free (real) variable is
equivocal, or has an indeterminate meaning, we do not agree with
him. ‘ Red (x) * is a proper sentence with a perfectly unambiguous
meaning; it is exactly equivalent in meaning to the sentence
(occurring in our Language II and in Russell’s language)
‘(x)(Red(x))’.
The expression which arises out of a given expression by
the substitution of 3 i for 3 i will be designated syntactically by
*^^(3 )* exactly defined in the following manner.
The positions in at which occurs freely in are called the
substitution-positions for 3^ in 2ti; Slj
is that expression which
arises out of when 3^ is replaced by 3i at all the substitution-
positions in ; here 3i niust be so constructed that no variable
which is bound at any of the substitution-positions for 3^ in
occurs as a free variable in 3i- 9Ii, 31 does not occur
as a free variable, then
designates the unchanged ex-
pression
Example: Let 0i, 65, 06 represent the previously mentioned sen-
tences, and let 3i be the variable ‘x’ and 32 ‘y*. Then 0j (tm')
represents the sentence : ‘ Pj (0) v (jc) 5 (P, (x, 0')) v P3 (0) *. 05
is 06. — ‘(3 means: “For every number y, there
is a next higher number.” Here a 3» m which ' x* occurs as a
free variable, for instance ‘x’’, must not be substituted for ‘y’;
‘ (3 x) {x = 5f**) * is obviously false.
§ 7. The K-Operator
An expression of the form (K3) 3 (®) is not a sentence — as are
the corresponding expressions which have a universal or an
existential operator — it is a numerical expression. The K-operator,
(K3)3) is not a sentential operator but a descriptional operator;
§ 7 * THE K-OPERATOR 23
or, more specifically, a numerical operator. (K3i)3i(Si) means:
“the smallest number up to (and including) 3i for which Si is
true, and, when no such number exists, 0”. Examples: Let
‘Gr(a,b)’ mean: “a is greater than b*’; ‘(Ka:) 9 (Gr(jc,7))’ is
equivalent in meaning to ‘8’; ‘(K«)9 (Gr(jc,7)«Prim(jc))* is
equivalent in meaning to ‘0*.
In general, it follows from the meaning stated that two sen-
tences of the forms (i) and (z) below mean the same:
Pti [(K3 i)3i (prs(3i))] (i)
[~(33i)3i (pr»(3i))«pti(0)]v(33i)3i [Pt2(3i)*
(3*) 3i (~ (3* = 3i) 3 ~ Pt2 (3»))‘ Pti (3i)] (2)
The previously mentioned designations : ‘ operator-variable
‘limiting-expression*, ‘operand*, ‘bound* and ‘free* variables,
are also applied to expressions having K-operators. [In contrast
with the usual (Russellian) description, description by means of
the K-operator is never either empty or equivocal; it is always
univocal; hence in the use of the K-operator no special pre-
cautionary rules are necessary in our system. ]
§ 8 . The Definitions
Symbols for which no definitions are framed are called un»
defined or primitive symbols. The logical primitive symbols of
Language I consist of the eleven individual symbols mentioned
already (see p. i6), together with nu and all the 3. As descriptive
primitive symbols any pr^ or fuj, may be set up. All other 33,
pr, and fu, which it is desired to employ, must be introduced by
definitions, A 33 or a pr is always explicitly defined ; an fu either
explicitly or regressively.
An explicit definition consists of one sentence; a regressive
definition of two sentences. Each of these sentences will have the
form: 3i = 32» ©1 = 62* expression 3i (or Si) is called
the definiendum, and contains the symbol which is to be defined.
32 (or ©2) called the definiens.
In an explicit definition, the symbol which is to be defined
occurs only in the dehniendum ; in a regressive definition, on the
other hand, it occurs also in the definiens of the second sentence.
24
PART I. THE DEFINITE LANGUAGE I
For the rest, a definiens may contain only either primitive symbols
or such as have previously been defined. Thus the order in which
definitions are set up may not be altered arbitrarily. To each
defined symbol belongs a chain of definitions^ by which is meant
the shortest series of sentences which contains the definition of
that symbol together with the definitions of all defined symbols
occurring in the chain. The chain of definitions of a symbol is
always finite, and (apart from the order of succession) uniquely
determined.
To the explicit definitions^ in the wider sense in which the word is
used here, belong both the explicit definitions in the narrower sense
— that is to say those where the definiendum consists only of the
new symbol (for instance, the definition of a 35 in I) — and the
so-called definitions in usu — those where the definiendum contains
other symbols besides the new symbol (for instance the definition
of a pr or an fu in I).
The definition of a numeral, 331, has the form: 33 i = 3 -
The definition of a predicate, pti”, has the form :
The explicit definition of a functor fui" has the form :
fUi"( 3 i. 3 *.- 3 n) = 3 - [Example: Def. i, p. 59.]
The regressive definition of an fui** has the form: {a) fUi"
(nu, 32 , • • • 3 n) = 3 i ; (*) K” ( 3 i', 32, • • • 3 n) = 32 - In 32 . fUi" is always
followed by the argument-expression 3 i, 32 • • • 3 n» variables of
which are not bound. , [Example: Def. 3 for ‘prod*, p. 59; the
first equation serves for the transformation of fUi(nu,3); the
second equation refers fUi(33*» 34) to fui (33i 34) so that, for
example, in ‘prod(6,y)*, by using the second equation six times
and the first equation once, ‘prod* may be eliminated.] Further,
every definition-sentence must fulfil the following two condi-
tions: (i) in the definiens, no free variable may occur which does
not already occur in the definiendum; and (2) two equal variables
must not occur in the definiendum.
If condition (i) is not made, then it is possible for definitions to be
framed by means of which a contradiction may be inferred. This
may be shown by an example (Lesniewski gives a similar example
for the sentential calculus [Neues System], pp. 79 f.). We define
apr‘P*:
25
§8. THE DEFINITIONS
P(*) = (Gr(jc,y).GrCy,s)) (i)
(1) (Gr(7,6).Gr(6,s))DP(7) (2)
Gr(7,6).Gr(6,5) (3)
( 2 ) (3) , P.(7) (4)
(i) P(7)3(Gr(7,4).Gr(4,s)) (5)
(5) P(7)DGr(4,5) (6)
(6) ~Gr(4,s)D~P(7) (7)
~Gr(4,s) (8)
(7) (8) ''P(7) (9)
(4) and (9) contradict one another.
It is not necessary, on the other hand, to make the converse
condition ; a variable not present in the definiens may be present in
the dehniendum. (Compare, for instance: Def. 3.1, p. 59.)
Condition (2) is made, not for the purpose of avoiding contra-
dictions, but for the purpose of assuring retranslatability. For in-
stance, if one defined : ‘ P (jc, jc) = Q {x) \ then ‘ P * in ‘ P (0, 1) * could
not be eliminated.
•
If we have a sentence of the form 3i = 32> Si = S2, then,
as we shall see later, 3i n^ay be replaced by 32> ®2»
and conversely, in every other sentence (p. 36). TKis means that
every explicitly defined symbol, wherever it occurs, may be
eliminated by the help of its definition. But in the case of a re-
gressively defined symbol, the elimination is not always possible.
\Example: If ‘prod occurs in a sentence in which "x* is free
(e.g. ‘ prod {x^y) = prod ( x) ’), then ‘ prod * cannot be eliminated. ]
We are now in a position to define more exactly the terms
* descriptive' and * logical' ^ which, up to the present, have only
been roughly explained.
If a symbol Qj is undefined, then Qi is called descriptive (Ob)
when is a pr or an fu. If Ui is defined, then is called an Uo
when an undefined Qb occurs in the definition-chain of Qj. An
expression is called descriptive (9lb) when an Qb occurs in ?Ii.
Qi is called logical (aj) when Qi is not an Qb- 9ti is called logical
(3li) when 9li is not an 3Ib.
§ 9. Sentences and Numerical Expressions
We will now name a few kinds of expressions. The most im-
portant of these are the sentences (S), and the numerical ex-
pressions (3). Hitherto we have given only inexact explanations
of them by reference to meanings. Now, however, these kinds
must be defined formally and exactly. We have already surveyed
26
PART I. THE DEFINITE LANGUAGE I
all the possible ways of constructing sentences and numerical
expressions in Language I, so that we have now only to enumerate
the various forms arising out of them.
An 6 may contain other S and 3 ^ parts; similarly a 3
contain other 3 » ^d also (by means of the K-operator) 6 as well.
Hence the definitions which we are about to give of the terms
‘sentence’ and ‘numerical expression’, to which we shall add the
auxiliary term ‘argument-expression*, refer to one another, but
only in so far as in determining whether a particular expression
submitted, is an S or a 3> shall refer to the question
whether a certain proper partial expression of is an 0 or a 3*
Thus it follows that this process of reference comes to an end
after a finite number of steps; the definitions are unambiguous
and do not contain a vicious circle. [Definitions of a strictly
accurate form will be given later within the framework of the
symbolically formulated syntax.]
A symbol of I which is either nu, or a defined numeral, or a 3,
is called a numeral (33). An expression of I is called an accented
eiqpression ( 0 t {Strichausdnickf) when it has one of the following
forms: i. nu; 2. 0 t'. [An 0 t is, therefore, either ‘ 0 ’ (improper
accented expression) or ‘0’ with one or more accents ‘
An expression of 1 is called a numerical expression ( 3 ) when it
has one of the following forms: i. 33; 2. 3' \ 3- fu" (^Irg") ; 4. (K31)
3 i (S), where 31 does not occur as a free variable in 3 i* Regres-
sive definition for ‘n-termed argument-expression’ (^rg**) in
I : an is a 3; an has the form 9 lrg", 3-
An expression of 1 is called a sentence (0) when it has one
of the following forms: i, 3 = 3 (‘Equation’); 2. pr"(9lrg”);
3. '^(S); 4. (S)ucrftt(0); 2li(S), where has the form
(3i) 3 nr (3 3i) 3i» and where 31 does not occur as a free variable
in 3i- [R is not necessary for the operator-variable to occur in
the operand as a free variable ; if not, then (0^) is equivalent in
meaning to 0^.]
The most important classification of expressions is the classifica-
tion into sentences and non-sentences. The frequent division of
expressions which are not sentences into expressions with ‘inde-
pendent meaning’ (“proper names” in the wider sense) and the rest
(“incomplete”, “unfulfilled”, “ synsemantic ” expressions) maybe
regarded as more significant from the psychological than from the
logical standpoint.
§ lO. CONCERNING TRANSFORMATION RULES
27
B. RULES OF TRANSFORMATION FOR
LANGUAGE I
§ 10. General Remarks Concerning
Transformation Rules
For the construction of a calculus the statement of the trans-
formation rules, as well as of the formation rules, as given for
Language I, is essential. By means of the former we determine
under what conditions a sentence is a consequence of another
sentence or sentences (the premisses). But the fact that S2 is a
consequence of Si does not mean that the thought of Si will be
accompanied by the thought of S2. It is not a question of a
psychological but of a logical relation between sentences. In the
statement of Si, the statement of S2 is already objectively in-
volved. We shall see that the relationship which is here indicated
in a material way can be purely formally conceived. \Example:
Let Si be *(jc)5 (Red(jc))*, and S2 ‘Red (3)*; given that all
positions up to 5 are red, then it is also given (implicitly) that the
position 3 is red. In this particular case, perhaps, S2 will have
been “thought ” simultaneously with Si ; but in other cases, where
the transformation is more complicated, the consequence will not
necessarily be thought coincidently with the premisses.]
It is impossible by the aid of simple methods to frame a de-
finition for the term ‘consequence’ in its full comprehension.
Such a definition has never yet been achieved in modern logic (nor,
of course, in the older logic). But we shall return to this subject
later. At present, we shall determine for Language I, instead of
the term ‘ consequence ’, the somewhat narrower term ‘ derivable \
[In constructing systems of logic, it is generally customary to use
only the latter narrower concept, and it is not usually clearly
understood that the concept of derivability is not the general
concept of consequence.] For this purpose, the term 'directly^
derivable^ will be defined, or — as it is more commonly expressed —
rules of inference will be laid down. [S3 is called ‘directly de-
.rivable ’ from Si or from Si and S2> when, with the help of one
of the rules of inference, S3 can be obtained from Si, or from Si
and S2.]
28
PART I. THE DEFINITE LANGUAGE I
By a derivation with the premisses Si , ©2 , . • . Sm (of which the
number is always finite, and may also be 0), we understand a
series of sentences of any finite length, such that every sentence
of the series is either one of the premisses, or a definition-sentence,
or directly derivable from one or more (in our object-languages I
and II, at most two) of the sentences which precede it in the
series. If S„ is the final sentence of a derivation with the pre-
misses Si, ... S,n» f^hen Sn is called derivable from ©!,...©,„•
If a sentence when materially interpreted is logically univer-
sally true (and therefore the consequence of any sentence what-
soever), then we call it an analytic (or tautological) sentence.
[Example: ‘Red(3)V /^Red(3)*; this sentence is true in every
case, independently of the nature of the position 3.] But this
is another concept that is not amenable to formal analysis by
means of simple methods, and it will be discussed later. First,
we propose to give the definition of the somewhat narrower
term demonstrable’. [This is the usual procedure; Godel was
the first to show that not all analytic sentences are demonstrable.]
Si is called demonstrable when Si is derivable from the
null series of premisses, and hence from any sentence whatso-
ever.
If a sentence when materially interpreted is logically invalid,
we shall call it contradictory, [Example: ‘Red(3)« ^Red(3)’;
this sentence is false in every case, independently of the nature
of the position 3.] We shall return later to the consideration of
this concept. For thfe moment, we shall take, instead, the some-
what narrower term ‘ refutable ’. Si is called refutable when at
least one sentence ^ S2 is demonstrable, S2 being obtained from
Si by the substitution of any accented expressions for* all the 3
which occur as free variables. [Examples: ‘Prim(:c)’ is refutable
because ‘ ^Prim( 0 i*")* is demonstrable.] A closed sentence Si
is thus refutable when, and only when, Si is demonstrable.
A sentence is called synthetic when it is neither analytic nor
contradictory. A sentence is called irresoluble when it is neither
demonstrable nor refutable. This last term is somewhat more
comprehensive than the term ‘synthetic*. We shall see later that
every logical sentence is either analytic or contradictory; and that
therefore synthetic sentences are only to be found amongst the
descriptive sentences. On the other hand, in I — and likewise in
§ lO. CONCERNING TRANSFORMATION RULES 29
every sufficiently rich language — there are logical sentences that
are irresoluble. (Compare § 36.)
For reasons of technical simplicity, it is customary not to
formulate the entire system of rules of inference, but only a few
of these, and in place of the rest to set up certain sentences which
are demonstrable (on the basis of the total system of rules), the
so-called primitive sentences. The choice of rules and primitive
sentences — even when a definite material interpretation of the
calculus is assumed beforehand — is, to a large extent, arbitrary.
Often a system can be changed (without changing the content) by
omitting a primitive sentence, and, in its place, laying down a
rule of inference — and conversely.
We also shall lay down rules of inference (that is to say, the
definition of * directly derivable ’) and set up primitive sentences
for our object-languages. In this method, a derivation with certain
premisses is to be defined as a series of sentences of which each
one is either one of the premisses, or a primitive sentence, or a
definition-sentence, or is directly derivable from sentences which
precede it in the series. A derivation without premisses is called
a proof. A proof is thus a series of sentences of which each is
either a primitive sentence, or a definition-sentence, or is directly
derivable from sentences which precede it in the series. The final
sentence of a proof is called a demonstrable sentence.
§11 . The Primitive Sentences of Language I
We shall give here, not the individual primitive sentences, but
a series of schemata of primitive sentences. Each schema will
determine a kind of sentence to which an unlimited number of
sentences belong. For instance, by means of the schema PSI i
it is determined that every sentence of the form Si D('^ 013 02)
is to be called a primitive sentence of the first kind, where 0i
and 02 may be sentences which are constructed in any way
whatsoever. [It is customary to lay down primitive sentences
instead of schemata, and in Language II we also shall use that
method. But for that purpose, variables for 0 , pr, and fu are
necessary. For example: the primitive sentence PSII i (p. 91)
corresponds to the schema PSI i. But, because in Language I
we have not the necessary variables at our disposal, we cannot
30
PART I. THE DEFINITE LANGUAGE I
construct the primitive sentences themselves, but only schemata.
The sentences which are here called primitive sentences of the
first kind are, in II, indirectly demonstrable sentences. They
follow from PSII i by substitution.]
Schemata of the Primitive Sentences of Language I
(a) Primitive sentences of the so-called sentential calculus,
PSIl. 0 iD(^SiDS2)
PSl 2 . (^SiDSi)D0i
PSI 3 . (SiD(5j)o[(<5*d<38)3(®i3S3)]
(b) Primitive sentences of the sentential operators (limited).
PSI 4 . (3i)rai(S0=SiQi)
PSI 5 . (3i)3*' (Si)= [(3 i)32 (Si).
PSI6. (33i)3*(Si)=~(3i)32(~Sx)
(c) Primitive sentences of identity,
PSI 7 . 3i=3i
PSI8. (3i=3*)o[SiOSx(^)]
{d) Primitive sentences of arithmetic.
PSI 9 . -~(nu= 3 ')
PSIio. (3i'=32')3(3i = 3a)
(e) Primitive sentences of the K-operator,
(3*)3i[~(33=3i) 3 ~Si . ®,] )
We shall now see that all primitive sentences when materially
interpreted are true, and (in the case of PSI5-11) that by the
substitution of any 33 for the free 3, true sentences follow from
them.
For PSI 1-3 : this is easy to show by means of the truth-value
tables (on p. 20). For PSI 4: the two sides of the equivalence
§ II. THE PRIMITIVE SENTENCES OF LANGUAGE I 3 1
are equal in meaning according to the meaning already given
for the limited universal operator — therefore both are true or
both are false. For PSIs: when something is true for every
number up to w+ 1, then it is also true for every number up to n
and for w + 1, and conversely. For PSI 6 : “ There is a number in
the series up to n having such and such a property**, is equivalent
in meaning to the sentence: “It is not true for every number up
to n that it does not possess the property in question.** PSI 4 and
5 represent, so to speak, the regressive definition of the limited
universal operator. PSI 6 represents the explicit definition of the
limited existential operator. While explicitly defined symbols can
always be eliminated, it is not always possible to eliminate re-
gressively defined symbols (compare p. 25). In like manner, a
limited imiversal operator cannot be eliminated when the
limiting expression contains a free variable (as, for example, in
PSI s). Limited universal operators and regressively defined fu
are not mere abbreviations, and if we were to renounce them,
the expressive capacity of the language would be very consider-
ably diminished. On the other hand, to renounce the limited
existential operator, the K-operator, and the symbols of con-
junction and implication together with all explicitly defined 33,
pr, and fu, would only succeed in rendering the language more
clumsy without in the least diminishing the extent of the ex-
pressible.
The symbol of identity or equality ‘ = * between 3 is here in-
tended (as in arithmetic) in the sense that 3i = 82 is true, if and
only if 3 i and 32 designate the same number, to use the common
phrase. From this, it follows that PSI 7 and 8 are valid. PSI 9
means that zero is not the successor of any other number, and is
therefore the initial term of the series. PSI 10 means that different
numbers have not the same successor. PSI 9 and 10 correspond
to the fourth and third axiom respectively in Peano*s system of
axioms for arithmetic. The material validity of PSIii follows
from the meaning of the K-opeutor previously given (§7).
32
PART I. THE DEFINITE LANGUAGE I
§12. The Rules of Inference of Language I
63 is called directly derivable in I from Si(Rl i, 2), or from
Si and 02(^1 3 > 4 )t when one of the following conditions RI 1-4
is fulfilled:
Rl I. (Substitution.) S3 has the form Si
Rl2« (Junctions.) (a) S3 is obtained from Si by replacing a
partial sentence (proper or improper) of the form S4V Ss by
S4DS6, or conversely;* (A) likewise with the forms 04* S5 and
^(/-^S4V /^Ss); (c) likewise with the forms S4 = 05 and
(643S*)-(S6DS,).
jRl3. (Implication.) 02 has the form Si D 03.
Rl4« (Complete induction.) Si has the form S3
has the form SsDSa
, and S2
That which we here formulate in the form of a definition of
‘ directly derivable * is usually formulated in the form of rules of
inference. Thus, the conditions just stated would correspond to
the following four rules of inference :
1. Rule of substitution. Every substitution is allowed.
2. Rule oi junctions, {a) A partial sentence S4 V S5 can always
be replaced by S4D S5, and conversely. Correspondingly with
(A) and (c).
3. Rule of implication. From Si and SiDSs, S3 may be
deduced.
4. Rule of complete induction. Example: From pri(nu) and
pri(3i)3pti(3i')» pri(3i) may be deduced.
These rules are formulated in such a way that, when the sen-
tences are materially interpreted, they always lead from true sen-
tences to further true sentences. In the case of RI i, this follows
• (NotCy 1935.) In the German original, RI 2(a) relates to 640 Sg
and ^ 04 V 65, and replaces a definition of the implication symbol.
Dr. Tars^ has called my attention to the fact that, instead of this,
the above form of RI 2(a), which stands for a definition of the dis-
junction symbol, must be taken, because in PSI 1-3 the implication
and not the disjunction symbol is employed. For the same reason
PSII 4 has also been changed (see § 30).
§ 12. THE RULES OF INFERENCE OF LANGUAGE I 33
from the interpretation of the free variables given earlier in the
book, and in the case of both RI2 and 3, from the truth-value
tables (p. 20). RI 2 represents, so to speak, an explicit definition
of the symbols of disjunction, conjunction, and equivalence,
which merely serve as abbreviations. RI4 corresponds to the
ordinary arithmetical principle of complete induction: if a pro-
perty belongs to the number 0, and if this property is an hereditary
one (that is, one which, if it belongs to a number n, also belongs
to n + i) then it also belongs to every number (Peano’s fifth
axiom).
§ 13. Derivations and Proofs in Language I
That a certain sentence is demonstrable, or derivable from
certain other sentences, will be shown by giving a proof or a deri-
vation. We shall find the more fruitful method to be that of
proving universal syntactical sentences which mean that all sen-
tences of such and such a form are demonstrable, or (ferivable from
other sentences of such and such a form. Sometimes the proof of a
universal syntactical sentence of this kind can be effected by the
construction of a schema for the proof or the derivation. The
schema states how the proof or derivation can be carried out in
individual cases. Another fruitful method, which in many cases
obviates the construction of special schemata, is based on the fact
that universal syntactical sentences about demonstrability or
derivability can be inferred from other sentences of the same kind.
That is, if S3 is derivable from S2, and S2 from Si, then S3 is
also derivable from Si*, for this derivation can be obtained by
placing the first two derivations one after the other. If Si is
demonstrable, and S2 derivable from Si, then (^2 is slso de-
monstrable. Further, if SiD02 i® demonstrable, then S2 is de-
rivable from Si (according to RI 3). The converse is not always true,
but only the following ; if Si is closed and S2 derivable from Si,
SiDS 2 is demonstrable. [The counter-example for an open Si is
as follows: let Si be ‘:c = 2’, and S2 be ‘(ai:)3 (^ = 2)*; S2 is de-
rivable from Si (Si and S2 are false); but Si 302 this case is
not demonstrable and is even false, for S2 results from this sen-
tence by the substitution of * 2 * for ‘ x *, and by the application of
RI3]
4
34
PART I. THE DEFINITE LANGUAGE I
We will give simple examples of a proof-schema and a deriva-
tion-schema together with several universal syntactical sentences
about demonstrability and derivability. [The references on the
left-hand side of the page to primitive sentences and rules are
only there to facilitate understanding — they do not belong to the
schema. On the other hand, the special conditions stated in
words, to which a particular expression is subjected (for instance,
@3 in the derivation schema below), are essential to the schema. ]
Example of a Proof-Schema
PSIi SiD(-SiDSi) (i)
PSI2 (-SiDSi)dSi (2)
PSI3, in which the sentence ^SiDSi will be taken for Sj
and taken for Sg :
(S,D(~SiDSi))d([(~SiD< 9 x)DSJ 0 [SiDSJ) ( 3 )
(I) (3) RI 3 ((-SxDSi)3S03(Si3Si) ( 4 )
(*) ( 4 ) RI 3 SpSi (5)
Theorem i3.i« is always (that is, for any sentential
design ®2) demonstrable.
We shall designate the syntactical theorem No. n of § m by
‘Theorem m#n*. The syntactical theorems 13. 1-4 refer to that
part of Language I which corresponds to the so-called sentential
calculus. This part comprises PSI 1-3 and RI 1-3.
Theorem 13.2. ®iV/^®i is always demonstrable. This is the
so-called principle of excluded middle.
Theorem 13.3. ®i and ®i are mutually derivable.
Theorem 13.4. If ®i is refutable, then every sentence ®2
is derivable from ®i . — Since Qi is refutable, a demonstrable
sentence ^ ®3 exists such that ®3 is obtained from ®i by means
of substitution.
Thus, in addition to ®i, we can use
~63 as a
premiss in the derivation schema :
(I)
( 2 )
(I) RIi
®S
(3)
PSI I
S3D(<~S3DS2)
(4)
(3) (4) RI 3
~ 63063
( 5 )
( 2 ) (5) RI 3
(0)
§ 13. DERIVATIONS AND PROOFS IN LANGUAGE I 35
The syntactical theorems which follow refer to that part of the
language which goes beyond the calculus of sentences — namely,
to the calculus of predicates, [This is usually known as the
functional calculus. For the most part, up to the present, the term
‘ predicate ' has been applied only to the one-termed pr. ] In this
domain Language I deviates further from the usual form of
formal language (Russell and Hilbert). Since Language I is a
language of co-ordinates, the method of complete induction
(RI 4) will often be applied in the proofs and derivations.
A. Syntactical Theorems about Universal Sentences
Theorem 13.5. Every sentence of one of the following forms is
demonstrable :
(«) (3i)3i(Si)3Si(^;);
(c) (3 i)3i(®i)-®i» provided ji does not occur as a free
variable in Si-
Theorem 13.6.
(«) (®i) always derivable from (3i)3i(Si):
(d) (3) 3 (Si) is always derivable from ;
(c) Sj is always derivable from (3i)3i(®i)» provided 3^ does
not occur as a free variable in Si ;
(‘O (3i)3i(Si)d(3i)3i(® 2) is always derivable from
( 3 i) 3 i(SiDS*);
(«) ( 3 i) 3 i(< 5 i) = (3i)3i(S 4) is always derivable from Si = 63
(this follows from Theorem 6 6, d),
B. Syntactical Theorems about Existential Sentences
Theorem 13.7. The following sentences are always demon-
strable :
(«) (33i)nu(S0 = Si(^3^);
W (3 3i) 3i' (®i) = [(3 3x) 3i (Si) V )] ;
36
PART I. THE DEFINITE LANGUAGE I
W ei(^Jo(33,)3i(<Si).
Theorem I3«8. ( 33 i) 3 i(®i) is derivable from Si; and if 3i
does not occur as a free variable in Si, then the converse is also
true. — Further relations of derivation analogous to Theorem 6
may be stated.
C. Syntactical Theorems about Equations
Theorem I3«9« The following sentences are always demon-
strable :
(.)(3.=3.).[3.(|)=3.(|J].
(6) (3i=32)3(3*=3i);
W [(3i=3*).(a=33)]3(3i=33)-
Theorem 13.10. The following sentences are derivable from
D. Syntactical Theorems about Replacement
Theorem 13.11. ?Ii 32 ^ is derivable from 3 i = 3 ^ and 9 li 3 i 3 l 2 >
provided the latter is a sentence. In other words: if an equa-
tion is assumed, then in any sentence, the left-hand term of the
equation may be replaced by the right-hand term (and, similarly,
the right-hand term by the left-hand term).
Theorem 13.12. 9liS23l2 is derivable from Si = S2 and 3IiSi2l2,
provided the latter is a sentence. In other words: if an equi-
valence is assumed, then, in any sentence in which the second (or
the first) equivalence-term occurs, it may be replaced by the first
equivalence-term (or the second, respectively). The proof is ob-
tained by means of the analysis of the different forms in which one
sentence can occur in another one (compare, for instance. Theorem
6c). [Compare also Hilbert [Logik] p. 61 ; the condition that the
same free variables must occur in Si and S2 is not necessary in
our form of language.]
The difference between replacement and substitution: In the case
§ 13 . DERIVATIONS AND PROOFS IN LANGUAGE I 37
of substitution, all expressions of the same kind (namely, the free
variables) which occur in the sentence must be transformed
simultaneously; on the other hand, in the case of a replacement,
no attention need be paid to the remaining parts of the sentence.
The possibility of presenting the defmitions in the form of
equations depends upon Theorems ii and 12 (compare § 8). On
the basis of an explicit definition, the definiendum can in every
case be replaced by the definiens, and conversely.
§ 14. Rules of Consequence for Language I
The case may arise where, for a particular ptj, say pti, every
sentence of the form pri(St) is demonstrable, but not the uni-
versal sentence pri(3i). We shall encounter a pr of this kindjater
on (§ 36). Although every individual case is inferable, there is no
possibility of inferring the sentence pti (3i). In order to create that
possibility, we will introduce the term ‘ consequencfe which is
wider than the term ‘derivable’, and, analogously, the term
‘analytic’, which is wider than ‘demonstrable’, and the term
‘contradictory*, which is wider than ‘refutable’. The definition
will be framed so that the universal sentence in question, pri(3i),
although not demonstrable, will be analytic.
For this purpose it is necessary to deal also with classes of
sentences. Hitherto we have spoken only oi finite series of sen-
tences or of other expressions. But a class may be of such a
nature that it cannot be exhausted by means of a finite series. (It
may then be called an infinite class ; a more exact definition of this
term is unnecessary for our purpose.) A class of expressions is
given by means of a syntactical determination (either definite or
indefinite) of the fonn of the expressions. For instance, by every
schema of primitive sentences an infinite class of sentences is
definitely determined. To speak of classes of expressions is only
a more convenient way of speaking of syntactical forms of ex-
pressions. [Later on, we shall see that ‘ class * and ‘ property ’ are
two words for the same thing.]
We shall apply the following designations (of the syntax-
language) to classes of expressions (sentences for the most part).
‘A’ {Klasse) will be the general term. will be taken to
represent the class of which the only element is ; ‘ ( 9 Ii , . . . ’,
PART I. THE DEFINITE LANGUAGE I
38
the class consisting of the elements 9Ii,9l2,
sum of the classes ^2* ^ expressions is called a
descriptive class when at least one of the expressions in it is a
descriptive expression; otherwise it is called a logical class. (In
this Section, ‘Ai’ and so on always designate classes of sentences.)
Si is called a direct consequence of fti (in I) when one of the
following conditions, DC i, 2, is fulfilled :
DC 1. Ai is finite, and there exists a derivation in which RI 4
(complete induction) is not used and of which the premisses
are the sentences of 5ti and the last sentence is Si;
DC 2. There exists a 5i such that is the class of
all sentences of the form ®^y»
5^2 is called a direct consequence-class (in I) of Ri when every
sentence of R2 is a direct consequence of a sub-class of R^, A finite
series of (not necessarily finite) classes of sentences, such that
every class (except the first) is a direct consequence-class of the
class which directly precedes it in the series, is called a consequence-
series (in 1). 01 is called a consequence (in I) of 5 ti, when there
exists a consequence-series of which Ri is the first class, and
{©i} the last. S„ is called a consequence of ©j, or of ©j, ©j, . . . Sn»>
when ©„ is a consequence of {©1}, or of {Si,©2, ••• Sm}» re-
spectively.
In rule DC i we are not obliged to exclude rule RI 4 (complete
induction). But its additional application would be superfluous
since, on the basis of the definitions given, it can be shown that
©3 is always a consequence of |©3 j
class be ftj. Then, as is easily seen, every sentence of the form
(st) from Ri and is therefore, according to DC i,
a direct consequence of Ri; thus, the class of these sentences, Rg*
is a direct consequence-class of Ri ; and according to DC 2, S3 is
a direct consequence of Rj* ^md therefore a consequence of R^.
Theorem I4.i. If a sentence is derivable from other sentences,
then it is also a con^quence of them.
§ 14. RULES OF CONSEQUENCE FOR LANGUAGE I 39
The consequence-relation has a wider extension than the
derivability-relation. The rule DC 2 could, as the above ex-
position shows, be partially replaced by RI4. A complete
equivalent of DC 2 is impossible to obtain either by means of
RI4 or any other rules of inference of the former kind, that is to
say, any rules concerned with the concept ‘directly derivable'.
For, since a derivation must consist of a finite number of sen-
tences, these rules always refer to a finite number of premisses.
But DC 2 in general refers to infinite classes of sentences. [Com-
pare the example given at the beginning of the section. pri(3i)
is not a consequence of any proper sub -class of the class of
sentences pri(St), still less a consequence of a finite sub-class.]
Thus we have now two dijferent methods of deduction: the more
restricted method of derivation^ and the wider method di the
consequence-series, A derivation is a finite series of sentences; a
consequence-series is a finite series of not necessarily finite
classes. In the case of derivation, every individual step (i.e. the
relation ‘directly derivable*) is definite, but not the relation
‘derivable*, which is defined by the whole chain of derivations.
In the case of the consequence-series, the single step (i.e. the
relation ‘ direct consequence *) is already indefinite, and therefore
all the more the relation ‘ consequence *. The term ‘ derivable * is
a narrower one than the term ‘consequence*. I’he latter is the
only one that exactly corresponds to what we mean when we say :
“This sentence follows (logically) from that one**, or: “If this
sentence is true, then (on logical grounds) that one is also true.’*
In the usual systems of symbolic logic, instead of the concept
‘consequence*, the narrower but much simpler concept ‘de-
rivable* is applied, by laying down certain rules of inference.
And, in fact, the method of derivation always remains the funda-
mental method; every demonstration of the applicability of any
term is ultimately based upon a derivation. Even the demonstra-
tion of the existence of a consequence-relation — that is to say,
the construction of a consequence-serics in the object-language —
can only be achieved by means of a derivation (a proof), in the
syntax-language.
A sentence is called anal3rtic (in I) when it is a consequence
of the null class of sentences (and thus a consequence of every
sentence) ; it is called contradictory when every sentence is the
40
PART I. THE DEFINITE LANGUAGE I
consequence of Si; it is called L-determinate when it is either
anal3rtic or contradictory ; it is called synthetic when it is neither
analytic nor contradictory.
A sentential class Ri is called analytic when every sentence of
Ri is an analytic sentence ; contradictory when every sentence is a
consequence of and synthetic when it is neither analytic nor
contradictory.
Two or mote sentences are called incompatible (with one
another) when the class consisting of them is a contradictory class ;
otherwise they are called compatible.
Theorem 14.2. Every demonstrable sentence is analytic;
every refutable sentence is contradictory. The converse is, how-
ever, not universally true.
Theorem 14.3. Every Sr (and Mr) is either analytic or contra-
dictory. Only an Sb (or a i\b) can be synthetic.
Proof: I. Let Si be a closed Sr- The application of the rules of
reduction which are to be given later (§340) leads, in a finite
number of steps, either to ‘nu = nu’, or to the negation thereof.
Here, every reduction-step is in agreement with DCi. There-
fore Si is L-determinate.
2. Let us assume that every Sr in which n different free
variables occur is L-determinate ; we will show that, in that case,
the same is true for every Si with w + 1 free variables. Let Si be
an Si with the n + 1 free variables 31, 32 > • • • 3 n> 5n+i- Consider the
class Ri of the sentences of the form Si j • Every one of these
sentences contains n free variables, and therefore, according to
our assumption, it is L-determinate. Then according to DC 2,
Si is a direct consequence of 5 li. Now, either all the sentences of
ill ^re an^ilytic or at least one of them is contradictory — say S2.
In the first case. Si is also analytic; in the second case, Si is
contradictory because S2 is a direct consequence of Si- There-
fore Si is L-determinate and consequently every Si with «+i
free variables is L-determinate.
3. By the Principle of Induction it follows from (i) and (2)
that every Si is L-determinate.
Example: Fermat's Theorem:
‘ (Gr (*, 0) . Gr {y, 0) . Gr {z, 0) . Gr («, 0"))d
.~(8um [pot {x, u), pot {y, u)] = pot (z , «)) ’
§ 14. RULES OF CONSEQUENCE FOR LANGUAGE I 4 1
(the definitions of ‘ Gr *, * sum \ ‘ pot * will be given in § 20) is a
logical sentence and therefore, according to Theorem 1I4.3, is
certainly either analytic or contradictory. Up to the present it is
not known which of these two is the case.
Theorem 14.4. A ftj is contradictory if, and only if, at least
one sentence belonging to it is contradictoiy. But a may
be contradictory without any of the sentences belonging to
it being contradictory. (For this reason it is important that not
only the sentences, but also the classes of sentences, should be
classified as analytic, contradictory, or synthetic.)
Example: Let pti be an undefined prt ; then the sentences ptj (tiu)
and '^pri(nu) are synthetic; but the class of these two sentences
(like their conjunction) is contradictory.
By means of the concept ‘ analytic *, an exact understanding of
what is usually designated as ‘ logically valid * or ‘ true on logical
grounds’ is achieved. Hitherto it has for the most part been
thought that logical validity was representable by the term
‘demonstrable’ — that is to say, by a process of derivation. But
although, for the majority of practical cases, the term ‘demon-
strable ’ constitutes an adequate approximation, it does not exhaust
the concept of logical validity. The same thing holds for the pairs
‘demonstrable’ — ‘analytic’ and ‘refutable’ — ‘contradictory’, as
for the pair ‘derivable’ — ‘consequence’.
In material interpretation, an analytic sentence is absolutely
true whatever the empirical facts may be. Hence, it does not state
anything about facts. On the other hand, a contradictory sentence
states too much to be capable of being true ; for from a contra-
dictory sentence both every fact and its opposite can be deduced.
A synthetic sentence is sometimes true — namely, when certain
facts exist — and sometimes false; hence it says something as to
what facts exist. Synthetic sentences are the genuine statements
about reality.
If we wish to determine what a sentence Sj (in the material
mode of speech) means, without leaving the domain of the formal
to go over into that of the material interpretation of the sentence,
we must find out what sentences are the consequences of that
sentence. Among these sentences we may ignore those which are
the consequences of every sentence — ^that is to say, the analytic
sentences. The non-analytic consequences of Si constitute the
42 PART I. THE DEFINITE LANGUAGE I
whole domain of that which is “ to be got out ’* of Sj. We therefore
define as follows ; by the logical content of 0 ^ or (in I) we
understand the class of non-analytic sentences (of I) which are
consequences of 0 ^ or fti respectively (in I). The “content” or
“sense” of a sentence is often spoken of without determining
exactly what is to be understood by the expression. The defined
term ‘ content ’ seems to us to represent precisely what is meant by
‘ sense ’ or * meaning * — so long as nothing psychological or extra-
logical is intended by it.
We call sentences or classes of sentences having the same
content equipollent. Two sentences are obviously equipollent
when and only when each of them is a consequence of the
other.
Ilf discussions as to whether certain sentences have the same
sense (or meaning), objections of the following nature are very often
made to the logician: “But this sentence and that cannot have the
same sense (or meaning) because they are connected with quite
different thoughts, images, and so on.*’ To this objection it may
be replied that the question of logical congruence of meaning has
no connection with that of the agreement of conceptions and the
like. The latter is a question of a psychological nature and must
therefore be decided by empirical and psychological investigation.
It has nothing whatsoever to do with logic. (Furthennore, the
question as to what ideas are connected with particular sentences is
a vague and ambi^ous one; the answer will differ according to the
person who is the subject of experiment and to the particular
circumstances.) The question whether two sentences have the same
logical sense is cohcerned only with the agreement of the two sen-
tences in all their consequence-relations. The concept of ‘having
logically the same sense ’ is thus adequately expressed by the above-
defined syntactical term ‘equipollent*. The concept of two terms
having the same meaning j which will be comprehended by the syn-
tactical term ‘ sjmonymous *, is an analogous case.
Theorem 14.5. Mutually derivable sentences are equipollent.
The converse is not universally true.
Two expressions, 'ilj and are called s3monymous when each
sentence 0 ^ in which occurs is equipollent (not, for example,
merely equal in truth-value) to that sentence 02 which arises out
of 01 when ?Ii is replaced by ®y nieans of this concept
‘synonymous*, the relation which is designated in the material
mode of speech as that of ‘ having the same meaning *, is formally
comprehended.
§ 14. RULES OF CONSEQUENCE FOR LANGUAGE I 43
Examples: ‘2*, ‘ 0 “*, ‘i**, ‘sum (1,1)* are synonymous. Let ‘te*
be an undefined f Ub ; then, even when ‘ te (3) = 5 * is an empirically
true sentence, *te(3)’ is not synonymous with ‘5*, and, more
generally, not with any 331 or St. [But ‘ te (3) ’ is synonymous with
‘ 5 ' in relation to ‘ te (3) = 5 * ; on this point see § 65.] In the English
language ‘Odysseus* and ‘the father of Telemachus* are not
synonymous, even though they both designate the same person.
Theorem 14.6. If 3 i = 3-2 analytic, then 3 i and 32 are
synonymous, and conversely.
Theorem 14.7. (a) If 0 iDS 2 is analytic, then ©2 is a conse-
quence of Sj. {h) If 02 is a consequence of ©j, and ©^ is closed,
then ©iD© 2 is analytic.
Proof of 76. (7 a follows naturally from DC i and RI 3.) Let ©1
be closed. We will state the proof for general cases ; for the special
case where ©j is logical, the proof is a considerably simple^ one.
We will call ©2 an analytic implicate of ©^ if ©jD ©2 is analytic.
(A) Every primitive sentence of I is an analytic implicate of ©j.
(B) If ©3 and ©4 are both analytic implicates of ©1, and if ©5
is directly derivable from ©3 and ©4, then ©5 is also an analytic
implicate of ©1.
(C) It follows from A and B that : if ©3 is an analytic implicate
of ©1, and ©4 is derivable from ©3 without the application of
complete induction, then ©4 is also an analytic implicate of ©j.
Therefore: if ©3 is an analytic implicate of ©^ and if, according
to DC I, ©4 is a direct consequence of ©3, then ©4 is also an
analytic implicate of ©j.
(D) If, according to DC 2, ©4 is a direct consequence of ili
and ii every sentence of 5^^ is an analytic implicate of ©j, then ©4
is also an analytic implicate of ©j.
(E) It follows from C and D that : if every sentence of ft2 is an
analytic implicate of ©1, and if ©4 is a consequence of R2» ©4
is also an analytic implicate of ©|.
(F) Since ©^ is an analytic implicate of itself, therefore the
following holds: if ©2 is a consequence of ©4, then ©2 is an
analytic implicate of ©1.
llieorem 14*8. Two sentences are synonymous when a^id only
when they are equipollent. [This is valid for Languages I and II
and for certain other languages also. Compare Theorem 65.4^.]
Theorem I4«9. If ©^ = ©2 is analytic, then ©1 and ©2 are
equipollent, and conversely.
44
PART I. THE DEFINITE LANGUAGE I
From Theorems 6, 8, and 9, it follows that the definiendum
and the definiens of a definition-sentence are synonymous.
Remarks on terminology. Instead of the expression ‘analytic*,
Wittgenstein [Tractatus'\ — and, following him, the literature of the
Vienna Circle up to the present time — uses the expression ‘tauto-
logical* or ‘tautology* (which, however, is only defined for the
sentential calculus). On the other hand, it is customary to apply
the term ‘ tautological * to transformations of sentences — namely, to
those which do not enlarge the content. We say, for example : “ The
inferences of logic are tautological.** It is a matter of experience,
however, that the use of the word ‘tautological* in these two
different senses, especially as the first does not correspond to the
usual mode of speech, easily leads to misunderstanding and con-
fusion. It would seem, therefore, more practical to retain the ex-
pression in the second case only (‘tautological conclusion*) and
to adopt the expression ‘ analytic * to apply to the first case (‘ analytic
sentences *). This term, which was used in the first place by Kant,
has t>een more sharply defined by Frege {^Grundlageri\ p. 4). He calls
a sentence analytic when, for its proof, only “the universal logical
laws** together with definitions are necessary. Dubislav [Ana~
lytische] has pointed out that the concept is a relative one ; it must
always be referred to a particular system of assumptions and methods
of reasoning (primitive sentences and rules of inference), that is to
say, in our terminology, to a particular language.
The expression ‘ contradictory * (or ‘ contradiction *) was likewise
introduced by Wittgenstein (within the calculus of propositions).
In addition to the expressions ‘analytic* and ‘synthetic* Kant did
not use a third expression for the negations of analytic sentences.
It might be worth considering whether the expression ‘analytic*
should be taken as a generic term (according to the suggestion
of Dubislav [Analytische\, as opposed to ordinary usage) and then
‘analytically true* and ‘analytically false*, or ‘positively analytic*
and ‘negatively analytic*, used in place of ‘analytic* and ‘contra-
dictory*.
C. REMARKS ON THE DEFINITE FORM
OF LANGUAGE
§ 15. Definite and Indefinite
The form of language most commonly used in modern logic
is that which Whitehead and Russell [Princ. Math] have
built up on the foundations laid by the work of Frege, Peano,
Schroder, and others. Hilbert [Logik\ uses a different symbolism.
§ 15- DEFINITE AND INDEFINITE 45
but his form of language has remained the same in all essentials.
In choosing the symbols for our object-languages I and 11 , we
have adopted the symbolism of Russell, because it is the most
widely known. In the form of the language we follow the main
outlines of the system of Hilbert and Russell, but we deviate from
it in some essential points, especially in our Language I. The
most important deviations are the following: the use of symbols of
position, instead of names of objects (language of co-ordinates);
limited operators (definite language); and two different kinds of
universality.
We have already spoken (§ 3) about the nature of our language
considered as a language of co-ordinates (symbols of position as
arguments). In this form of language there is an essential syn-
tactical difference between the situation-terms for positions, and
the other determinations by means of which any properties of
positions are stated. The latter we shall call qualitative terms. A
relation of situation in the simplest case will be expressed by means
of an analytic (or contradictory) sentence (e.g. “Positions 7 and
6 are neighbouring positions *'). On the other hand, a qualitative
relation, in the simplest case, will be expressed by means of a
synthetic descriptive sentence (e.g. “Position 7 arid position 6
have the same colour”). The foraier sentence is determined by a
logical operation, namely, a proof; the latter, on the other hand,
can only be decided on the basis of empirical observations, that is
to say, by derivation from observation-sentences. In this fact lies
an essential difference which is obliterated when the language is
so constructed — as by the methods hitherto accepted — that
situation-determinations and qualitative relations are expressed
in a syntactically identical manner.
We shall call a symbol of Languages I and II definite when
it is either an undefined constant or a defined one in the definition-
chain of which no unlimited operator occurs ; otherwise indefinite.
An expression will be called definite when all the constants which
occur in it are definite, and when .ill the variables in it are limitedly
bound ; otherwise indefinite.
All definite expressions are closed. In the case of the expres-
sions in Language I, the concepts ‘definite’ and ‘closed’ are
identical; similarly, ‘indefinite* and ‘open’. We call I a definite
language because, in I, all constants and all closed expressions are
46 PART I. THE DEFINITE LANGUAGE I
definite. [In the strictest sense, only a language in which all the
expressions are definite may be called a definite language.] [On
the admissibility of indefinite concepts, compare §§ 43-45.]
To “ calculate” a numerical expression, say 3 i> means : to trans-
form 3 i ii'to an St ; or, more exactly, to prove a sentence of the
form = To “resolve” a sentence, say Si, means: either to
prove or to refute it. Now it can be shown that every definite 3 i can
be calculated; and that every definite Si can he resolved. Moreover,
there exists a definite method by means of which this calculation
and resolution respectively can be achieved. This is the so-called
reduction which will be explained later. If pti is a definite pti^ and
fUi is a definite fui”, then pti (Sti, ... Stn) is always resoluble, and
fUi(Sti, ...St„) is always calculable.
§ 16. On Intuitionism
Some of the tendencies which are commonly designated as
* finitist * or ‘ constructivist * find, in a certain sense, their realiza-
tion in our definite Language I. “In a certain sense”, let it be
noted; for inasmuch as these tendencies are, as a rule, only
vaguely formulated, an exact statement is not possible. They are
chiefly represented by Intuitionism (Poincar^ ; and in contemporary
thought, above all Brouwer ; also Weyl, Heyting, and Becker) and
allied opinions (for example, F. Kaufmann and Wittgenstein).
The points of contact will presently be stated precisely, but our
own view differs: from the tendencies in question in one essential
respect. We hold that the problems dealt with by Intuitionism can
be exactly formulated only by means of the construction of a
calculus, and that all the non-formal discussions are to be re-
garded merely as more or less vague preliminaries to such a
construction. The majority of the Intuitionists, however, are of the
opinion that a calculus is something inessential, a mere supple-
mentary appendix. Only He3rting has made an interesting at-
tempt towards formalization from the standpoint of Intuitionism
— ^we shall say something about his method later.
Once the fact is realized that all the pros and cons of the In-
tuitionist discussions are concerned with the forms of a calculus,
questions will no longer be put in the form: “What is this or
that like ? ” but instead we shall ask : “ How do tve wish to arrange
§ 1 6. ON INTUITIONISM 47
this or that in the language to be constructed?** or, from the
theoretical standpoint : “ What consequences will ensue if we con-
struct a language in this or that way?**
On this view the dogmatic attitude which renders so many
discussions unfruitful disappears. When we here construct our
Language I in such a way that it is a definite language, and thus
fulfils certain conditions laid down by Intuitionism, we do not
mean thereby to suggest that this is the only possible or justifiable
form of language. We shall, on the contrary, include the definite
Language I as a sub -language in the more comprehensive
Language II, and the form of both languages will be looked upon
as a matter of convention.
In Language I, all pr^ and fur are definite; the question whether
a definite pri can be attributed to a definite number or not, or
whether a definite fui has a definite value for a definite number or
not, is always resoluble. This fact corresponds to the Intuitionist
requirement that no concept be admitted for which a. method of
resolution is not stated. Further, the non-application of unlimited
operators in I has the result that unlimited universality, although
it can be positively expressed (namely, by means of free variables),
cannot be negated. We can only say, either : * P (x) *, which means :
“All numbers have the property P**; or: ‘ '^P(;c)*, which means:
“All numbers have the property not-P**, “No number has the
property P.” On the other hand, “Not all numbers have the
property P** is not expressible in I; in II it will be expressed by:
‘ (P(jc))*. This sentence will be treated in II (as in the lan-
guages of Hilbert and Russell) as equivalent in meaning to
‘(3^) ('^P(^))*» which means: “There is (at least) one number
which has the property not-P.** In I there are no such unlimited
existential sentences, and this fact also corresponds to a condition
laid down by Intuitionism, namely that an existential sentence
may only be stated if either a concrete example can be produced,
or, at least, a method given by the aid of which an example can be
constructed in a finite, limited number of steps. For the In-
tuitionists, existence without rules for construction is considered to
be “inadmissible** or “nonsensical** (“meaningless**). It is not
quite clear, however, whether (and within exactly what limits),
according to their point of view, existential sentences, and perhaps
even negated universal sentences also, should be excluded by
48 PART I. THE DEFINITE LANGUAGE I
means of syntactical rules of formation, or whether only certain
possibilities of transformation should be excluded. The issue in-
volved is, above all, the question of indirect proof by means of
the refutation of a universal sentence.
Let us take an example : (let ‘ P ’ be a pti) :
(*)(P(*))(Si). ~(*)(P(*))(S*), (3*)(~P(*))(S3).
In classical mathematics (and therefore also in the logic both of
Russell and Hilbert, as well as in our II), when is reduced ad
absurdum, first S2 inferred, and then from it the existential
sentence S3. It is in order to exclude this inference leading to an
unlimited, non-constructive existential sentence that Brouwer re-
nounces the so-called Lou; of Excluded Middle. The language-form
of I, however, shows that the same result can be achieved by other
methods — namely, by means of the exclusion of the unlimited
operators. In I, Si can be translated into ‘P(jip)*, but S2 and S3
are not translatable into I. Here, the Law of Excluded Middle
remains valid in I (Theorem 13.2). The exclusion of this law, as
is well known, brings with it serious complications which do not
occur in I. Thus Language I fulfils the fundamental conditions of
Intuitionism in a simpler way than the form of language suggested
by Brouwer (and partially carried out by Heyting).
In I universality is expressed in two different ways: by free
variables, and by universal operators. Because the latter are
always limited in I, the two methods of expression are not of equal
value. We can make use of these two possibilities of expression in
order to express two different kinds of universality.
Let us consider some examples: i. All the pieces of iron on
this table are round.” 2 a. “ All pieces of iron are pieces of metal. ”
zh. “All pieces of iron are magnetizable.” In case i, the sen-
tence is dependent on an empirical test of a series of individual
instances; a sentence of this kind is only determinable in a limited
domain. Hence, the limited universal operator is best adapted to
formulate it. In cases 2 a and 2 unlimited universality occurs.
The validity of these sentences cannot be determined by the testing
of individual instances. Sentence 2 a is analytic and follows from
the definition of * iron ’. Sentence 2 b has (like all so-called laws
of nature) the character of a hypothesis. Such a sentence is de-
pendent upon the acceptance of a convention which in its turn is
§l6. ON INTUmONISM 49
dependent upon a partial testing of individual instances. The use
of free variables is adapted to the formulation of the unlimited
universality of the examples za and zb.
F. Kaufmann has rightly emphasized the difference between
the two kinds of universality (he designates them, in common with
Husserl, as individual ( i ) and specific (2 a) universality). [Whether
his criticism, based on this differentiation, of the logic of the
present time, especially that of Russell, and of the Theory
of Aggregates, is entirely justified, is not here considered.]
Perhaps the form of Language I represents the realization of a
part of Kaufmann ’s ideas, but it is not possible to decide this
point exactly, since Kaufmann, like Brouwer, has laid down no
foundations for the construction of a formal system. A deviation
from the language-form of Language I consists in the fact that
Kaufmann, like Wittgenstein, considers sentences of the type 2 h
to be inadmissible, since they are neither analytic nor limited, and
in consequence cannot be completely verified in any way. In
contrast with this view, the language-form of 1 also admits
synthetic unlimitedly universal sentences.
§ i6a. Identity
The following explanations are concerned with the symbol
‘ = * considered as the symbol of identity in the narrower sense
(that is to say, as used between 3 or between object-designations)
and not as the symbol of equivalence (that is to say, as used be-
tween S). The symbol of identity occurs in Languages I and II
(as also in the languages of Frege, Behmann, Hilbert) as an un-
defined symbol. Following Leibniz, Russell defines ‘ x =y * in the
following way: “jc and y agree in all their (elementary) pro-
perties.’* Wittgenstein rejects the symbol altogether and suggests
a new method for the use of variables by which it may be avoided.
Philosophical discussions concerning the justification of these
various methods seem to us to be wrong. The whole thing is only
a question of the establishment of a convention whose technical
efficiency can be discussed. No fundamental reasons exist why
the second or third of these methods should not be used instead
of the first in Languages I and II. As it happens, the Leibniz-
Russell method is only applicable in Language II ; there the de-
50
PART I. THE DEFINITE LANGUAGE I
finitionm>uldtakethefoini( 3 i= 3 ^=(pi) (Pi(3i)3Pi(5t)). Against
this definition the objection is sometimes raised (for instance on
the part of Wittgenstein, Ramsey, Behmann) that it is at least
conceivable for two different objects to coincide in all their
properties. But this objection is dismissed as soon as ** all pro-
perties” are understood as including those of position. That is
already true even for name-languages, and most certainly true for
co-ordinate languages : 3i designate the same place when
every property of position which holds for 3i holds also for 3*-
It would in any case be sufficient in the definition instead of * all pro-
perties ' to say * all properties of position’ (for which, for instance,
instead of p a sort of variable limited to pr(, say pi, could be used).
Wittgenstein’s criticism goes still further: he does not merely
reject Russell’s definition, but refuses to make use of the symbol
of identity at all. But it seems to us that all that emerges from his
remarks about this symbol is that sentences of the form ^ are
— at least in the simplest cases — ^not synthetic, but analytic: it
does not seem to us to follow that such sentences are altogether
inadmissible. In order to avoid the use of the symbol of identity,
Wittgenstein proposes to use a rule of substitution which differs
from the one usually employed both in mathematics and in logic.
His rule is that, for different variables, different constants must be
substituted. The shorter form, ‘P(jif,y)\ of Wittgenstein’s lan-
guage corresponds to the usual form of sentence ‘ ^ (jc =y) 3 P («, y )’.
On the other hand, ‘ P (»,y) V P (*, «) ’ corresponds to the sentence
‘P(jip,y)’. Since'Wittgenstein does not formulate any new rule of
substitution but only states a number of examples, it is not clear
how he intends to carry out his method. A closer examination
shows that his method of variables leads to certain complications.
Hence it seems to us to be better to retain the ordinary use of
the symbol of identity and with it at the same time the ordinary
rule of substitution.
According lo Wittgenstein’s idea, ‘ P (0, 0) *, for instance, cannot
be derivable from * P (x, y) ’. But if, by a derivation step, * P (0, y) * is
obtained from 'P(af,y)’, then it is not possible to see why in the
derived sentence *0’ may not be substituted for ’y’. Hence in
order to prevent this substitution at some later stage in the deriva-
tion, a special expedient must be introduced by writing scmiething
of this sort: *^’^P(0,y)’; and for this purpose suitable new rules
must be laid down.
§l6fl. IDENTITY 51
Russell’s use of the symbol of identity for the definition of
finite classes by the enumeration of their elements is equally re-
jected by Wittgenstein. In our opinion, however, there is no need
to reject these classes, but only to observe the difference (certainly
an important one) subsisting between them and those classes which
are defined by means of properties in the narrower sense. This
is effected by means of suitable syntactical differentiations; the
essential point is the difference between the prj (and in particular
the finite definite prj) and the ptb-
§ 17. The Principle of Tolerai^ce in Syntax
In the foregoing we have discussed several examples of
negative requirements (especially those of Brouwer, Kaufmann,
and Wittgenstein) by which certain common forms of lan^age
— methods of expression and of inference— would be excluded.
Our attitude to requirements of this kind is given a general
formulation in the Principle of Tolerance: It is not our business to
set up prohibitions^ but to arrive at conventions.
Some of the prohibitions which have hitherto been suggested
have been historically useful in that they have served to emphasize
important differences and bring them to general notice. But such
prohibitions can be replaced by a definitional differentiation. In
many cases, this is brought about by the simultaneous investiga-
tion (analogous to that of Euclidean and non-Euclidean geo-
metries) of language-forms of different kinds — for instance, a
definite and an indefinite language, or a language admitting and
one not admitting the Law of Excluded Middle. Occasionally it
is possible to replace a prohibition by taking into account the in-
tended distinctions within one particular form of language, by means
of a suitable classification of the expressions and an investigation
of the different kinds. Thus, for example, while Wittgenstein and
Kaufmann reject both logical and arithmetical properties, in I de-
scriptive and logical predicates !iave been distinguished. In II
definite and indefinite predicates will be distinguished and their
different properties determined. And further, in II, we shall
differentiate between limitedly universal sentences, analytic un-
limitedly universal sentences, and synthetic unlimitedly universal
sentences, whereas Wittgenstein, Kaufmann, and Schlick all ex •
52
PART I. THE DEFINITE LANGUAGE I
elude sentences of the third kind (laws of nature) from language
altogether, as not being amenable to complete verification.
In logic j there are no morals. Everyone is at liberty to build up
his own logic, i.e. his own form of language, as he wishes. All that
is required of him is that, if he wishes to discuss it, he must state
his methods clearly, and give syntactical rules instead of philo-
sophical arguments.
The tolerant attitude here suggested is, as far as special mathe-
matical calculi are concerned, the attitude which is tacitly shared
by the majority of mathematicians. In the conflict over the logical
foundations of mathematics, this attitude was represented with
especial emphasis (and apparently before anyone else) by Menger
([Intuitionismtis] pp. 324 f.). Menger points out that the concept of
constnictivity, which Intuitionism absolutizes, can be interpreted
both in a much narrower, and in a much wider sense. The im-
portance for the clarification of the pseudo-problems of philosophy
of applying the attitude of tolerance to the form of language as a
whole will become clear later (see § 78).
PART n
THE FORMAL CONSTRUCTION OF THE
SYNTAX OF LANGUAGE I
§ i8. The Syntax of I can be Formulated in I
Up to the present, we have differentiated between the object-
language and the syntax-language in which the syntax of the
object-language is formulated. Are these necessarily two separate
languages ? If this question is answered in the affirmative (as it is
by Herbrand in connection with metamathematics), then a third
language will be necessary for the formulation of the syntax df the
syntax-language, and so on to infinity. According to another
opinion (that of Wittgenstein), there exists only one l^guage, and
what we call syntax cannot be expressed at all — it can only “be
shown**. As opposed to these views, we intend to show that,
actually, it is possible to manage with one language only; not,
however, by renouncing syntax, but by demonstrating that with-
out the emergence of any contradictions the syntax of this
language can be formulated within this language itself. In every
language S, the syntax of any language whatsoever — whether of
an entirely different kind of language, or of a sub-language, or
even of S itself — can be formulated to an extent which is limited
only by the richness in means of expression of the language S.
Thus, with the means of expression of our definite Language I,
the definite part of the syntax of any language whatsoever — for
instance, of Russell’s language or of Language II, or even of
Language I itself — can be formulated. In the following pages,
the latter undertaking will be carried out — ^that is to say, we shall
formulate the syntax of I — as far as it is definite — in I itself. In this
process it may happen that a sentence Si of I, when materially
interpreted as a syntactical sentence, will say something about Si
itself, and without any contradiction arising.
We differentiate between descriptive and pure syntax (see pp. 6f .).
A sentence in the descriptive syntax of any language may state, for
instance, that an expression of such and such a kind occurs in
a certain series of positions. [A symbol occupies a position, an
54 part II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
expression occupies a series of positions.] Example: “On page 33
line 32 of this book, an expression of the form ‘3 = 531* occurs
(namely, ‘jc = 2*).** Since Language I has sufficient means of ex-
pression at its command for the purpose of describing the pro-
perty of a domain of discrete positions, a descriptive-syntactical
sentence of this kind may be formulated in I no matter whether
it describes an expression of another language or an expression
of I itself. It would, for instance, be possible to proceed by in-
troducing in I undefined pr^ for the different kinds of symbols of
the expressions to be described (later, we shall instead set up a
single undefined fUb, namely *zei’ \Zeichen^ \ for example, the ptt
‘ Var* for the variables, the ptb ‘LogZz’ for the logical numerals,
the ptb ‘ Id ’ for the symbol of identity, and so on. Let us now
designate the position on page 33 at which ‘;» = 2* begins, by ‘a*.
Then the aforementioned descriptive-syntactical sentence can be
formulated in I in the following manner:
‘Var (a). Id (a*). LogZz (a»)’.
This is a synthetic descriptive sentence. We can then, further,
define the ptb ‘ LogSatz * so that ‘ LogSatz (x, u) * means : “ In the
series of positions extending from jc to jc + m, an Sj occurs.** Then
the sentence: “Every expression of the form 3 = 331 is an ®i** will
be rendered in I by
‘ (Var (jc) • Id (jci) • LogZz (:di))D LogSatz {x, 2) ’ ;
this is an analytic sentence which follows from the definition of
‘LogSatz *.
§ 19. The Arithmetization of Syntax
As we have already mentioned, it is always possible to replace
any ptb by an fUb- Several different ptb may be called homogeneous
if at most one of them can appertain to any position. Then it is
always possible to replace a class of homogeneous ptb by one fUb,
by correlating one value of the fUb, either systematically or arbi-
trarily, to each one of the individual ptb- [Example: Let the class
of colours which are to be expressed be finite. We can ex-
press every colour by a ptb, ‘Blue’, ‘Red*, and so on. These ptb
are then homogeneous and therefore we can replace them by a
single fUb> say ‘col*, by numbering the colours in some way, and
§ 19- THE ARITHMETIZATION OF SYNTAX 55
Stipulating that *col(a) = b’ shall mean: *The position a has the
colour No. b.’] Similarly, in the formulation of the syntax of I
in I, we shall not designate the different kinds of symbols by
different ptb (as, for instance, in the example given in § i8 by
‘Id’, etc.) but by one fUb* namely ‘zei’. We shall correlate the
values of ‘zei* to the different symbols (symbol-designs), partly
arbitrarily and partly in accordance with certain rules. These
values are called the term-numbers of the symbols. For instance,
we shall co-ordinate the term-number 15 to the symbol of identity.
This means that (instead of ‘Id(a)’) we shall write ‘zei(a)=i5’
when we wish to express the fact that the symbol of identity
occurs at the position a. Not only the economy in primitive
syntactical concepts, but other reasons which will be discussed
later, justify the choice of this method of the arithmetizat§on of
syntax. (In this arithmetization, we make use of the method which
Godel [Unentscheidbare] has applied with such success in meta-
mathematics or the syntax of mathematics.)
In general, the establishment of term-numbers for the different
symbols can be effected arbitrarily. All that must be provided
for is the fact that, for the variables, of which the number is un-
limited, an unlimited number of term-numbers must be available
— likewise for the 33, pr, and fu. We will now specify infinite
classes of numbers for the term-numbers of these kinds of symbols
in the following way. Let p run through all the prime numbers
greater than 2. Stipulations: the term-number of a 3 shall be a^
(that is, a prime number greater than z); the term-number of a
defined 33 shall be a p* (that is, the second power of some prime
number greater than 2); the term-number of an undefined pr
shall be a p*; that of a defined pr, a p^; that of an undefined fu,
a p® (and specifically, the term-number of ‘ zei * shall be 3®, which
is 243) ; and that of a defined fu, a p®. But not all the numbers of
the classes determined in this way will be used as term-
numbers: the choice of them will be determined later. To the
remaining symbols — namely, the undefined logical constants — ^we
assign (arbitrarily) other numbers, namely :
to the symbol: 0() ,
the term-number: 4 6 10 12 14 15 18 20 21 22 24 26 30 33 34.
[The last three symbols are auxiliary symbols which do not occur
56 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
in the expressions of the language itself; see p. 68 concerning
them. ]
When any empirical theory is formulated in I, then the de-
scriptive primitive symbols of this theory are added to the logical
primitive symbols of Language I. Likewise in the formulation of
the descriptive syntax; here ‘zei* is the only additional primitive
symbol. In the following construction of the system of syntactical
definitions, however, ‘zei’ will not at first be used. For, at this
stage, we are not concerned with descriptive but with pure syntax,
and in this there are no additional primitive symbolsy since pure
syntax is nothing more than arithmetic. Just as term-numbers
correspond to symbols, so series of term-numbers correspond to
expressions. For example the series, 3, 15, 4, corresponds to the
expression ‘ jc = 0 *. The concepts and sentences of pure syntax refer
now not to the series of symbols but to the corresponding series of
term-numbers. Thus they are arithmetical concepts or sentences.
The formulation of the syntax becomes technically simpler if
we go one step further with the method of the correlation of
numbers. We will lay down a rule by which, to every series of
term-numbers, one number — we call it the series-number of the
series — will be uniquely correlated. In this way we shall no
longer have to deal with series of numbers but only with single
numbers. The rule is expressed as follows : pi*=i • • • • • Pn*” is to
be taken as the series-number for a series which consists of n
term-numbers, where Pt(i = i to n) is the rth prime
number in the order of magnitude. [Example: The series 3, 15,
4, and with it the expression *x = 0 \ has the series-number
2* •3^® #5^.] Since the factorization of a number into its prime
factors is unique, the series of term-numbers in its original order
may be regained from a series-number, and thereby also the
language-expression to which the series-number is correlated.
[The rules stated earlier concerning term-numbers are in addition
— ^but not necessarily — so arranged that no term-number is at the
same time the series-number of any series.]
The method of the construction of series-numbers may be
repeatedly applied. For instance, to a proof as a series of sen-
tences, there corresponds, to begin with, a series of series-numbers.
In accordance with the method described we can then correlate a
series-series-number to this se*ries of series-numbers.
57
§ IQ. THE ARITHMETIZATION OF SYNTAX
By means of these stipulations about term- and series-numbers,
all the definitions of pure syntax become arithmetical definitions,
namely, definitions of properties of, or relations between, numbers.
For instance, the verbal definition of ‘ sentence ’ will no longer have
the form; ‘*An expression is called a sentence when it consists of
symbols combined in such and such a way”; but instead: ”An
expression is called a sentence when its series-number fulfils such
and such conditions”; or, more exactly: “A number is called the
series-number of a sentence when it fulfils such and such con-
ditions.” These conditions are only concerned with the kinds and
order of the symbols of the expression, that is to say, with the
kinds and order of the exponents of the prime factors of the series-
number. We shall thus be able to express them purely arith-
metically. All the sentences of pure syntax follow. from these
arithmetical definitions and are thus analytic sentences of ele-
mentary arithmetic. The definitions and sentences of syntax
arithmetized in this way do not differ fundamentally from the
other definitions and sentences of arithmetic, but only in so far as
we give them a particular interpretation (namely the syntactical
interpretation) within a particular system.
If this method of arithmetization is not applied, certain dif-
ficulties arise in the exact formulation of the syntax. For instance,
let us consider the syntactical sentence : ” Si is not demonstrable ”,
which means: ” No sentential series having Si as its final sentence
is a proof.” If the syntax is not arithmetized but, instead, as was
suggested earlier, is constructed by the help of ptb (‘ Var*, etc.), we
may interpret it as a theory concerning certain series of physical
objects, namely, the series of written symbols. In a syntax of that
kind, it is certainly possible to express: “There exists no actual
written proof for Si”, but the sentence concerning the non-
demonstrability of Si means much more, namely: “No proof for
Si is possible'^ In order to be able to express such a sentence
about possibility in the non-arithmetized syntax (no matter
whether it is physically interpreted or not), the syntax would
have to be supplemented by a theory (not empirical but
analytic) concerning the possible arrangements of any elements —
that is to say, by pure combinatorial analysis. It proves, how-
ever, to be much simpler, instead of constructing a new com-
binatorial analysis of this kind in a non-arithmetical form, to use
58 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
the arithmetic of the natural numbers which already contains
within itself the whole of combinatorial analysis (whether of a
finite or a denumerable number of elements). This is the most
important reason for the arithmetization of syntax. In the
arithmetized syntax, the sentence under discussion would run as
follows: There exists no number which is the series-series-
number of a proof of which 6^ is the final sentence.” We shall see
that it is possible to frame an arithmetical definition for that
property of a number which consists in its being the series-series-
number of a proof which has a given series-number as the final
number. Our sentence will then have the form: “There is no
number having such and such an arithmetical property.” This is a
purely arithmetical sentence. By the arithmetization we are en-
abled, without using new and complicated auxiliary methods, to
express even those syntactical concepts (such as derivability and
demonstrability) which are concerned with a determinate pos-
sibility.
§ 20. General Terms*
We shall now formulate the construction of the syntax of I
presented in I as a system of arithmetical definitions. Explana-
tions, which indicate the interpretation of the terms involved
as syntactical terms, are appended to the definitions (in small
print). For the sake of brevity, the explanations are often in-
exactly and incorrectly formulated. The exact presentation of the
syntax consists solely of the symbolically formulated definitions. A^l
the symbols which are used in these definitions are either among
the logical primitive symbols of I (compare p. 24) or are defined
in the following pages. The defined symbols are, specifically,
certain 331, pti, and fui. In the following definitions, we shall use
the letters *k\ "l\../z^ as 3. (Later on, in Language II,
*p\ will be used as sentential variables, which do not occur
in I.)
The first definitions (D 1-23) are of a general nature and are
applicable to the syntax of any language whatsoever.
* Key to the symbols defined in this section :
nf: successor (Nachfolger) prod; product (Produkt)
sum: sum (Sunrne) po, pot: power (Potenz)
§ 20. GENERAL TERMS
59
Di.
D2. !• s[im{ 0 ,y)=y
2. sum = nf (sum (x^y))
D 3 . 1. prod(0,3;)=0
2. prod (x'^y) = sum (prod {Xyy\y)
D4. I. po(0,jv)=0i
a. po (*',>») = prod (po (A,y),>')
D 5 . pot( 2 ?,^) = po(/e,Jc)
D6. 1. fak(0)=0i
2. fak (jc*) = prod (fak {x)y ).
Explanation, D 1-6: Explicit (D 1,5) or regressive (Dz, 3, 4, 6)
definitions are here given for six fui having the meanings : Successor
(to jc); sum (of x and y); product; power (‘pot (5^,3^)*: 'x^* in
ordinary symbols); factorial (compare p. 14). ‘po* is only an
auxiliary concept for * pot * ; it is necessary because we have stipulated
that the first argument-place is to be taken as that to which the re-
gression refers.
By means of the regressive definitions stated for ‘sum* and
‘ prod *, the ordinary fundamental laws of arithmetic (the commu-
tative, associative, and distributive laws) and, further, all the known
theorems of elementary arithmetic can be proved with the help
of RI 4 (complete induction).
Dy, I. 1=01; 2 . 2 = 1 '; ...lo. 10 = 9i; ... 34 . 34 = 331.
Explanation: There are as many defined 35 as we shall require.
Here, a decimal to several places is taken as one indivisible 33.
D8. Grgl(AC,3;) = (3M);c(x=sum(jv,M))
D9. Gr(*,3/)= (Grgl(4f,>>).
Dio. Tib (*, 3') = (3 u)*(*= prod (>>,«))
Key to the symbols defined in
fak: factorial (Fakultdt)
Grgl: greater or equal (grosser
Oder gleich)
Gr: greater (grosser)
Tib : divisible (teilbar)
Prim, pr, prim: prime number
(Primzahl)
gl : term-number (Gliedzahl)
Ing : length (Lange)
letzt: last (letzte)
reihe: series (Reihe)
this section (continued ) :
zus : composed (zusammengesetzt)
ers: replaced (ersetzt)
InA : in the expression (im
Ausdruck)
InAR : in the expressional series
(in der Ausdrucksreihe) '
AlnA : expression in the expres-
sion (Ausdruck im Ausdruck)
AlnAR: expression in expres-
sional series (Ausdruck in der
A usdrucksreihe)
60 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
D II. Prim (x) = (jc = 0 ) • ^ (jc = i ) • («) jp ((m = i ) V
(i/ = ^)V^Tlb(jc,tt)))
Explanation^ D 8-i i : These are four pri having the meanings :
x'^y\x>y\x\s divisible by 3; ; jc is a prime number (compare p. 13).
Di2. I. pr( 0 ,A?) = 0
2. pr («•, x) = (Ry ) X (Prim (y) • Tib (x^y) •
Gr (>'.?>•(».*)))
Explanation: pr(/t,jc) is the nth (in magnitude) prime number
contained as a factor in x,
D13. I. prim( 0 ) = 0
2 . prim («•) = (Km) nf[fak (prim (»))] [Prim(m)«
Gr (m, prim (n))]
Explanation: prim(«) is the nth prime number (according to
magnitude).
D14. gl(«,*) = (K>»)* ['^Tlb(a;,pot[pr(«,*),>'i])]
Explanation: gl(n, a;) is the nth term-number of the series with
the series-number x.
D15. lng(jc) = (Kn)jip(pr(ni,jip)= 0 )
Explanation: lng(A;) is the length (that is to say the number of
terms) of the series with the series-number x.
D i6. letzt (jc) = gl (Ing (jc), x )
Explanation: letzt (:c) is the last term-number of the series with
the series-number x,
D17. I. reihe (5) = pot (2, r)
2 . reihe 2 (r, t) = prod (reihe (r), pot (3,0)
3. reiheS (5, t^ u) = prod (reihe 2 (r, f ), pot (s, m) )
Explanation: reihe (f) is the series-number (2*) of a series of which
s is the only term-number; reihe2(5, t) is the series-number (2 ' .30
of a series of which the term-numbers are s and t\ and so on.
(In ‘reihe2* ‘2* is not a 33 but a component part of the indivisible
symbol *reihe2*.)
We will now introduce the following abbreviations for the
explanations. Instead of writing ‘term-number of...* we will
write / (for instance ‘TN negation symbol’, which is 21).
Instead of ‘series-number of...’ we will write (for in-
stance ‘SNOperjator* and so forth). Instead of ‘series-
series-number of...’ we will write ‘SSN_ » (for instance
§20. GENERAL TERMS 6l
‘SSNproQf*^ If rgad the verbal transcription of a definition
neglecting the indices, we shall get the syntactical interpretation of
the definition. (For instance in the explanation of D i8 : “ zus (x,y)
is the series which is composed of two partial series x and 3^''.)
On the other hand, if we read the transcription including the
indices, we shall get (usually in a form not literally accurate) the
arithmetical interpretation of the definition. (For instance, in the
case of D 18 : “ zus (x,y) is the series-number of the series which is
composed of two partial series having the series-numbers x
and y”.) In what follows we shall at first always work with the
indices but later on we shall use them only when it seems necessary
to do so for the sake of clarity.
Di 8 . 1. zus (x, y) = (Kx) pot [prim (sum [Ing (x), Ing (y)]),
sum (*,3/)] [(«)lng(*) (gl(«,2r) = gl(n,x)).(«)lng(3;)
(^(«=0)D[gl (sum [Ing (*),B],2r) = gl(n, >»)])]
2. zus 3 (x,y, z) = zus (zus (x,y), z)
3. ZU 94 (Xjy,z,u) = zus (zus 3 (Xf y,z),u)
and so on.
Explanation: zus (at, y) is the SN series which is composed of two
SN sub-series x and y (not : of ™ terms ; as different from ‘ reiheZ (r, t) *).
Correspondingly *zus 3 *, etc., in the case of composition from three
or more sn sub-series.
D19. ere {x, n,y)-(Ks!) pot [prim (sum [Ing (ar), Ing (>')]),
sum (*, 3/)] (0 u) x( 2 v)x [(x=zus 3 («, reihe [gl (w, *)],
»)).(« = 2us3 («,3', o)). (« = nf [Ing («)] )])
Explanation: er8(ac,n,y) is the SN expression, which follows from
the SN expression x when the nth SNterm in x is replaced by the
SN expression y.
D 20. InA {t,x) = (3 b) Ing (*) (<^ (b = 0 ) . [gl (b, *) = t] )
D 21. InAR (t,r) = (3 k) Ing (r) (~ (A = 0) . InA [/, gl (k, r )] )
D 22. AlnA (*,3') = (3 «)3’ (3 (3’ = zus 3 {u, x, v))
D 23. AlnAR (jc, r) = (3 k) Ing (r) (~ (ft = 0 ) . AlnA [*, gl (ft, r)] )
Explanation, D20: The tn symbol t occurs in the SN expression x.
D21: t occurs in an sn expression of the SSN expression-series r.
D 22 ; The expression x occurs (either as a proper or improper part)
in the expression y. D 23 : The expression x occurs in an expression
of the expression-series r.
62 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
§ 21 . Rules of Formation: (i) Numerical
Expressions and Sentences’***
D 24. einkl (x) = zusS (reihe (6), x, reihe ( 10) )
Explanation: If x is an sn then einkl (jc) is the sn bracketing of ac,
that is, the expression (^1).
D25. Var(j)= (Prim(5)«Gr(f,2))
Explanation: * * Var (5) ’ means that r is a prime number greater than
2 (thus, as a term-number it is a tn variable).
D 26. DeftZzl (s) = (3 ^ (V ar (m) • [j = pot (w, 2)] )
D27. DeftPradl (j), and D28. DeftFul(j), may be ana-
logously formulated.
Explanation, D 26-28 : 5 is a defined TN 35I (or prl, ful respectively)
when 5 is the second (or fourth or sixth respectively) power of a prime
number greater than 2. (Concerning the additional ' 1 * see later.)
Remark concerning the term-number of defined symbols
We have assigned as term-numbers to the defined S3rmbols of
the different kinds numbers of three classes — namely the second,
fourth, and sixth powers of prime numbers greater than 2. We
• Key to the symbols:
einkl : bracketing {Einklamme-
rung)
Var: variable (Variable) >
DeftZz, DeftPrAd, DeftFu: de-
fined numeral, predicate, func-
tor (definiertes Zahlzeichen,
Prddikat, Funktor)
UndPrad, UndFu : undefined . . .
(undefiniertes ...)
Zz: numeral (Zahlzeichen)
PrSd : predicate (Prddihat)
AOp, EOp, KOp, SOp: uni-
versal, existential, descrip-
tional, sentential operator
(All-y Eocistenz-, Satz-
Operator)
Op : operator (Operator)
ZA : numerical expression (Zahl-
ausdruck)
neg : negation (Negation) .
dis: disjunction (Di^unktion)
kon: conjunction (Konjunktion)
imp : implication (Implihation)
siq: equivalence (Aquivalenz)
Verkn: Junction (Verknilpfung)
gig : equation (Gleichung)
Satz: sentence
VR: variable-series (Variablen-
reihe)
UKstr : directly constructed
(unmittelbar k^truiert)
Konstr : construction (Konstruk-
tion)
KonstrA : constructed expres-
sion (konstruierter Ausdruck)
Geb : bound (gebunden)
Frei, Fr: free
Offen: open
Geschl: closed (geschhssen)
§21. NUMERICAL EXPRESSIONS AND SENTENCES 63
shall, however, later establish the method of defining symbols in
such a way that not all numbers of the three classes mentioned will
be used as term-numbers for defined symbols, but, instead; only
those numbers which fulfil certain conditions. We call a ™ symbol
hcLsedy when it either fulfils these conditions or is a primitive
symbol. These conditions will be formulated in such a way that
any symbol which fulfils them will refer back by means of its chain
of definitions to the primitive symbols. We call an expression
based when everyone of its ™ terms is a based term.
Those terms which will next be defined and of which the de-
signations (namely, the word-designation, the Gothic symbol, and
the predicate in the formal system) contain the additional ‘ 1 ’ or
‘2* (from ‘'defined 33I*', D26, to “constructed 2 *', D78) also
include symbols and expressions which are not based. These are
only auxiliary terms for the definitions which will follow later!
D29. UndPrad(s,n) = (g^)5[Var(^)#
(i = pot (prim[pot(/f,n)], 3) )]
Analogously D 30 : UndFu (s, «).
Explanation: 5 is an undefined (or fu") when a prime number
k greater than 2 exists, such that s is the third (or fifth) power of the
AE**th prime number. (This rule is laid down so that the position-
number n, which is essential for the syntactical rules, may follow
univocally from the term-number of a pr” or an fu".)
D 31. Zzl (s) = (DeftZzl (s) V Var (5) V (5 = 4))
Eocplanation: 5 is a tn when s is either a defined ™ 331 or else
a 3 or a nu (see p. 26).
D 32. Pradl (j) = [DeftPradl (j) V (3 n)s (UndPrad(j, n))]
Explanation: 5 is a when s is either a defined prl or an
undefined pr.
Analogously D 33 : Ful (5).
(That is, ful.)
034. AOpl (2, j, v)= [Var (5) • (a = zus (einkl [reihe (j)], u))
• ^InA(j, u)]
Analogously D 35 : EOpl (2, s^v); D 36 : KOpl (2, Sy v).
D 37. SOpl (2, Sy v) = (AOpl (2 , Sy v ) V EOpl (2, j, v))
• D 38. Opl (2, Sy v) = (SOpl (2, Sy v ) V KOpl (2, Sy v))
Edcplanationy D 34 : 2 is an ^^universal operator \ with the TNopera-
tor-variable s and the SNiimit v ; that is to say, 2 has the form (3^ ^1,
64 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
where does not occur in ^1. — D 35-D 38: existential operator!^
K-operatorlf sentential oper atari (that is to say» universal or
existential operator 1), operatorl (that is to say, sentential or
K-operatorl).
D 39. ZAl (jar) = (3 5) jar (3 v) ^ (3 w) jsr (3 J') jsr ( [Zzl (5) • [jar = reihe
(s )] ] V [jar = zus [v, reihe (14)] ] V [Ful (s) • (ar = zus [reihe (s),
einkl (a;)] )] V [KOpl (y, Sy v)m(z = zus [j/, einkl («;)] )])
Explanation: jsr is an SN^i^ when jar has one of the following forms :
35l» fwl(5Ia), (K3) ^1(513) (see p. 26). Here, ^1, and % are
any expressions whatsoever; on the other hand, in the case of a
32(D 53) is a 32| ^2 is a series composed of several 32 and
commas, and % is an S2. In contradistinction to a 32, a 3 (‘ ZA \
D 87) is based. Analogously in the case of Si (D 47), S2 (D 54),
and S(‘Satz’, D88).
D 40. neg (jc) = zus (reihe (21), einkl (ac))
D 41. dis (jc, y) = zus3 (einkl (jc), reihe (22), einkl (jy))
D42: kon (jc, y); D43: imp (jc, jy), and D44: aq (jc, 3;), are
analogous.
Explanation: If x and y are SNexpressions Wi, then neg(jjc) is
the ^^negation dis(af, y) the disjunction (^1) v (2I3); the cases
of coiyunction (kon), implication (imp), and equivalence (aq) are
analogous.
045. Verkn(ji[r,3', J3r)= [(ji[: = dis(jy, j3r))v(jx = kon(3/, 2r))v(2r = imp
(* = aq(j,a:))].
Explanation: x is an ^^junction of y and z : that is to say, x has the
form (^i)DcrIn(%|) where y is and z is
D4«. gig (*,>>) =2 us 3(*, reihe (i5),3')
Explanation: If x and y are expressions ^1, then glg(x,y) is
the equation 3li = 3I2.
D 47. Satzl {z) = (3 5) ar (3 v) ar (3 w) 2r (3 3;) ar ([jsr = gig (t;, w)]W
[Pradl (r) • (ar = zus [reihe (s), einkl (v)] )] V [jsr = neg (u)] V
Verkn (ar, u, w) V [SOpl (3;, r, v)m(z = zus [y^ einkl (w)] )])
Explanation: ar is an Si, when z has one of the following forms :
prK^tts). -W, or
(see p. 26). The difference between Si, S2, and S is analogous
to that between 3l> 32, and 3-
D48. VR (*, ») = ([lng (a:)i = prod (2, n)] . (*) Ing (*) (3 m) k
[(fe=0) V ([A = prod (2, m)'] . Var [gl (A, *)]) V ([A = prod
(2,m)].[gl(A,*) = i2])])
§21. NUMERICAL EXPRESSIONS AND SENTENCES 65
Explanation: An expression x is called an n-termed variable-series
when it consists of n variables and intervening commas.
D49. UKstrl (Zy «;)=([ZA 1 («;) • (cr = zus [w, reihe (14)])] V
[Satzl iw) • (a = neg («;))] V (3 n) Ing (w) (3 s) z [(VR (w, n) .
(Ful (5) V Pradl (j)) • (2: = zus [reihe (^), einkl (w)])) V
(VR (zVy n) • Var (5) • (2 = zus [«?, reihe 2 (12, s)]) )])
Explanation: An expression z is called directly constructed from
one expression w, say 5li, when it has one of the following forms :
I. ^li', where 3li is a 31 ; 2- '^(^1), where is an Si ; 3. ful(^i)
or prl(^i), where is a variable-series; 4. ^1,3, where is a
variable-series.
D 50. UKstr2 {zy Vy w) = [(3 s) z {2 y) ^ (ZAl (t>) • Satzl {w) .
Opl (3;, Sy v)%{z = zus [j, einkl (tt^)]))V (ZAl (t;) • ZAl (ti?) •
[;2r = glg (vy zi;)])v (Satzl (t?) • Satzl (zv) • Verkn (z, ty)) V
(3 n) Ing (®) (Var [gl («, ®)] . ZAl (w) . [» = ers (®, n, w)] )]
Explanation: An expression z is called directly constructed from
two other expressions Vy zv, say ^2» when it has one of the fol-
lowing forms: i. (3)^i(^a) or (33)^i(^) or (K3)^i(^2), where
is a 31 and an Si; 2. where and ^2 are 3l;
3. (^i)pcr!n ('412)1 where and ^2 are Si; or when, 4., z results from
if a 3 is replaced by UI21 where ^2 a 3h
D51. Konstrl (r)~{n) Ing (r) ['^(« = 0 )d ((3 s) r [Zzl (5) •
(gl («, r) = reihe (s ) )] V (g A) n (3 1 ) n [~ (A = 0 ) . ~ (/ = 0 ) .
(UKstrl [gl (n, r), gl (A, r)] V Ukstr 2 [gl (n, r), gl (A, r),
gl(Ar)])])]
Explanation: r is an ^^^constructionXy when r is an ssNseries of
SNexpressions of which each either is a 33 1 or is directly constructed
from one or two of the previous expressions occurring in the series.
(A series of this kind consists of 3^ and SI, or, more precisely, in
accordance with the following definitions, of 3^ and S2.)
D 52. KonstrAl (j£;) = (3 r) pot (prim [Ing (jp)], prod [Xy Ing (x)])
[Konstrl (r) • (letzt (r) = op)]
Explanation: An SNexpression x is called constructed! when it is
the last expression in an ssNconstructionl. [The limit for r results
from the following consideration. L#et r be the shortest ssNcon-
structionl of which the final SNsentence is x. Then lng(r)^lng(:»),
every prime factor of r is ^ prim (Ing ( op)), the number of these
factors is ^lng(jc), their exponents are ^x'y therefore
r ^ prim (lng(jip))**'”*^*\]
D 53. ZA 2 (x) = (KonstrAl (x ) . ZAl {x)")
SL
6
66 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
D 54. Satz 2 (x) = (KonstrAl (x) • Satzl (jc) )
Eocplanation: An expression jc is a 32 (or an 02 ), when it is both
constructed! and a 3l (or an 01, respectively); see explanation
of D 39.
D 55. Geb {Sy Xy «) = (3 t) x( 2 z)x (3 u) x{ 2 y)si (3 (3 w) ar
[(x = zus 3 {ty ZyU)^m(z = zus [yy einkl («;)] ) • Opl {yy 5, v) •
ZA 2 (v) • Satz2 (to) • Gr (fiy Ing (/)) • Grgl (sum [Ing (^),
lng(ar)],»)]
Eocplanation: The '^variable s is called hound in the SNexpression
X at the nth place (where the variable need not occur at this place)
if the following conditions are fulfilled : In a; an expression z of the
form ^i(^ occurs, where is an Operator 1 having a 32 as limit
and s as operator- variable ; ^2 is an 02 ; the nth place of x belongs
to z (see p. 21).
D 56. Frei(5,jc,n)= [Var(5)« (gl(n, jc) = 5)# /^Geb(f, A;,n)]
Explanation: The free variable s occurs at the nth place in jc.
D 57. Fr (s, x) = (3 fi) Ing (x) (Frei (f, x, n))
Eocplanation: s occurs as a free variable in x,
D 58. Offen (a:) = (3 i) a? (Fr (5, x)^
D 59. Geschl (x) = Offen (a:)
Explanation: x is open; x is closed (see p. 21).
§22. Rules OP Formation: (2) Definitions’*^
If a calculus is to contain definitions, then, under certain cir-
cumstances, there arises in it9 formulation a difficulty which is very
seldom taken into account. If all that is demanded of the defini-
* Key to the symbols :
VRDef: variable-series for the
definiens
DefZz, DefPr^d ; definition of a
numeral, predicate
DefexpFu, DeftexpFu: explicit
definition of a functor
DefrekFu, DeftrekFu: regres-
sive (rekursiv) definition of a
functor
Def , Df : definition sentence
(Definitionssatz)
Deft: defined (definiert)
Z: symbol (Zeichen)
UndDeskr: undefined descrip-
tive (deskriptiv) symbol
Undeft : undefined symbol
DefKette, DeftKette : definition-
chain (Definitionenkette)
Bas : based (hasiert)
Deskr: descriptive
Log: logical (logisch)
§22. FORMATION RULES : DEFINITIONS 67
tions admitted in the calculus is that they satisfy certain rules of
formation, the calculus will generally be a contradictory one.
Example: For instance, D i (p. 59) satisfies the formation rules
for definitions in I (§ 8). With the help of D i, the sentence
*nf(0) = 0i* is demonstrable. But the sentence ‘nf(:!c) = jc*** * is
likewise a definition of the admitted form and with its help the
sentence * '^(nf(0) = 0i)* is demonstrable. Thus, in I, sentences
which are mutually contradictory are demonstrable.
In order to avoid the contradiction, we usually make the addi-
tional requirement ‘‘that the symbol to be defined must not have
occurred in a definition which has already been framed*'. But a
requirement of this kind is a departure from the domain of the
calculus and of the formal method. In strictly formal procedure,
the decision as to whether a given sentence is an admissible
definition in a particular calculus or not is dependent solely upon
the form of the sentence and upon the formation-rules of the
calculus. But by virtue of the above non-formal requirement this
decision would become dependent upon the historical statement
as to whether certain sentences had been previously formulated or
not. And the same is true for the decision concerning the de-
monstrability of a given sentence (as our example shows). Now,
how can this difficulty be overcome ?
I. To begin with, it is obvious that the difficulty disappears if
in the formation of the language S in question, one of the following
procedures is adopted :
(a) No definitions at all are admitted in S.
{b) Only a finite number of particular definitions are admitted
in S, and these are ranged amongst the primitive sentences of S.
(c) Any number of definitions, for which rules of formation
are given, may be formulated in S. But the definitions are not
admitted in proofs; they are only admitted as premisses of
derivations. [Thus in the above example ‘nf(0)=0i* is not de-
monstrable, but only derivable from ‘ nf (jc) = *.] If a sentence Si
contains defined symbols (i.e. symbols based on certain definitions)
then, although it is not itself demonstrable, that sentence which
follows from Si as a result of the elimination of the defined symbols
is demonstrable.
Regressively defined symbols are not always eliminable. In a
68 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
definite language, in which the sentences of elementary arithmetic
(for instance: ‘prod (2, 3) = 6*) are to be demonstrable, an un-
limited number of regressive definitions, which must be employ-
able in the proofs, is necessary. Thus, for a language of this kind —
for instance. Language I — the above-mentioned ways out of the
difficulty are of no use. We shall have to discover some other
solution :
2. In Language I we shall allow an unlimited number of de-
finitions, including regressive ones ; but by means of suitable rules
we shall take care that from each defined symbol it is recognizable
how it is defined. This is possible in an arithmetized syntax. We
have previously established a class of numbers for the term-
numbers of the defined symbols of each of the three kinds, 33, pr,
and^fu; but, inside this class, so far, we have left the choice open.
Now, however, the rules to be laid down will determine this choice
in such a way that from the term-number of a defined symbol not
only its definition but also, indirectly, its whole chain of definitions
will follow univocally. In this way every so-called logical property
of any sentence — for instance, its demonstrability — becomes a
syntactical or formal property ; it depends solely upon the formal
structure of the sentence, that is, upon the arithmetical properties
of the term-numbers which constitute the sentence.
Rule for the choice of the term-numher of a defined symbol Qi in
Language I: In the definition of Oi, let Ui be replaced by a per-
manent auxiliary symbol as follows : a 33 by ‘ ^ ’ with the term-
number 30, a pr by * TT ’ with the term-number 33, an fu by ‘ ' with
the term-number 34. The definition which arises as a result of
this process then contains only old symbols ; thus its series-number
r— or in the case of the schema of a regressive definition, since it
consists of two sentences, its series-series-number r — can be
determined. Let us take as the term-number for Qi, when
^ 33 ^ Pty or an fu), the second (or fourth, or sixth,
respectively) power of the rth prime number. By applying this
rule the term-number for Ci is determined univocally; and con-
versely, from this term-number, r, and hence the definition schema,
and finally the definition, of ai, are univocally determinable.
By means of this rule, we can now establish the difference be-
tween based and non-based ™symbols. For instance, the fourth
power of a prime niunb^r p (greater than 2) is based (see p. 63)
§22. FORMATION RULES : DEFINITIONS 69
when p is obtained in the manner described from a definition schema
with the auxiliary symbol ‘tt* — ^ assuming that the analogous con-
dition holds for every defined TNsymbol occurring in the definition
schema. In order to formulate this condition, we shall later define
the concept of a chain of definitions (D 81). Before that, however,
it is necessary to define a list of auxiliary terms.
D 6o. VRDef y, n) = [VR (.x, n) . {k) Ing (x) (/) Ing (a?) ([Var
(gl {k, jf)) . (gl {K x) = gl (/, *))] = /)) . (i) y (Fr {s, y) o
liiA{s,x))]
Explanation: x is an n-termed sn variable-series which is suitable
(as argument-expression of the definiendum) for the SNdefiniens y
when the following is true: x is an w- termed variable-series; no two
equal variables occur in x\ every variable which occurs as a free
variable in y occurs in x also. (‘ VRDef * is an auxiliary term for the
purpose of abbreviation.)
D 61. DefZzl (ac) = { 2 z)x [(jc = gig [reihe (30), 2] ) • Geschl (2:) ]
Explanation: x is called an SN^efinitionl of a TNjj (that is to say,
an expression similar to the definition schema of a 35), when x has
the form J = tHi where is closed.
D 62. DefPradl (a?, n) = (3 w) a: ( 3 .t^) w (3 ^ [(w = zus [reihe (33),
einkl («;)] ) • (x = aq (w, 2:)) • VRDef (v, 2r, w)]
Analogously D 63: DefexpFul {x, n).
Explanation: x is called a definition! of a pr'* (or an explicit de-
finition! of an fu", respectively) when x has the form (^j) = ^2 (or
(^2li) = 9l2, respectively), where 3 li is an n-termed variable-series
which is suitable to
D 64. DefrekFul (r, n) = (3 x^) r (3 ^2) r (3 u^) x^ (3 v^) x^ (3 u^) X2
(3 ^2) *2 (3 s) *<2 (3 (3 »») « [(>■ = reihe2 (*i, *2)) . (x^ =
gig («i. *’1)) • 0*2 = gig («2. ®2)) • Var (r) . (() (Fr \t, v^) D
InA (t, «2)) . ik) Ing ([gl {K ^2) = 34] 3 W Ing (^) (~ (/ = 0 )
D [[gl [sum {k, 1 ), tJj] =gl (/, 3;)] . ~ Geb [gl [sum (k, 1 ), v^l,
2)2, sum (k, /)] ])).(« = ot') •([('” = 0 ) • (“1 = reihe 4 (34, 6, 4,
10)) . (m 2 = reiheS (34, 6, f, i-' to)) . (a = reiheS (6, s, 10))] V
(3 w) Ml [~ (m-O), (mj =zus [reihe (34), einkl (zus [reihe2
(4, iz), «)])]) . (m2 = zus [reihe (34), einkl (zus [reiheS (s, 14,
12), O’])]) . (3: = einkl (zus [reihe 2 {s, 12), w])) . VRDef (20,
»!, i«) . ~ InA {s, to) ])]
Explanation: r is called an ssNregressive definition! of an fu” when
r is a series of two expressions x^, X2 of the following kind. has
70 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
the form = jc, has the form ^14 = ^5; every variable which
occurs as a free variable in ^5 occurs in ^4 also; where the vari-
ables of 5le are not bound in ^5, occurs directly after every term
‘ <l> * which occurs in n is greater than 0 ; we put n = m+i. Now
there are two cases to be distinguished from one another. First
case: w = 0; then has the form <l>(x\u), ^4 the form and
^ the form (3i). Second case : m > 0 ; then has the form 0 (nu, ^),
2I4 the form (3i*, ^la), and ^ the form (3i,5l3); here ^3 is an
m-termed variable-series adapted to ^a and 3i does not occur in ^3.
[It is Wj: Vii u^: ^4,; v^: s: gzj^; z: w:
D 65. DeftZz 2 {t,y) = [DefZzl (3;) • (f = pot [prim {y\ 2] )]
Similarly D 66 : DeftPradl (/, n,3^) ; D67: DeftexpFu2(/,n,3^);
D 68 ! DeftrekFu 2 (/, r).
Explanation: t is a 33 (or pr", or fu”, respectively), which is
** defined2’* by means of the definition! y (or the explicit definition!
y, or the regressive definition! r, respectively).
D69. DefZz 2 {x, 0 = (3 y) ^ [DeftZz 2 (/, y) . (jc = ers [j, i,
reihe(0])]
Similarly D 70 : DefPrad 2 (x, n,t); D 71 : DefexpFu 2 (x, n, t).
Explanation^ D 65-68 : x is called a definition2 of a 33 f (or of a pr"
f, or an explicit definttion2 of an fu" t, respectively) when t is de-
fined2 by means of y, and x results from y when the first (or second
or first, respectively) TNterm, namely ‘ J * (or * tt * or ‘ respectively)
is replaced by the TNsymbol t.
D 72. I DefrekFu 2 (of, «, t) = (3 r) « (3 y) r [DeftrekFu 2 (f, 71, r) •
(gUi. »•)=>') • (* = crs[>», i,reihe( 0 ])]
Simdarly D 73 : 2 DefrekFu2 (ac, «, t).
Explanation: x is called the first (or second) SNpart of an
SSNregressive definition2 of an when the following conditions
are satisfied: t is regressively defined2 by means of the (regressive
definition!) r; y is the first (or second) part of r, and x results from
y when in y ‘ ^ * is replaced at the first place (or in all places at which
it occurs) by the TNsymbol L
D 74. Def 2 (jc, t) = [DefZz 2 (jc, t) V (3 n) Ing (x) (DefPrad 2
{x, n, i) V DefexpFu 2 (jc, n, t) V iDefrekFu 2 (jc, w, t) V
2DefrekFu2 (jc, w, <))]
• (NotCy 1935.) The stipulation that the variables of are free,
and the corresponding term of D64 “ '^Geb[gl[sum(^,/), Vj], V2y
sum(/i,/)]”, are obviously necessary, but they are omitted in the
German original (also in § 8). My attention was called to this
oversight by Dr. Tarski.
§22. FORMATION RULES : DEFINITIONS
Explanation: x is called a definition-sentence! of t when x is either
a definition 2 of a < or a pr or an explicit definition 2 of an fu f,
or the first or second part of a regressive definition 2 of an fu t,
D 75. I>eft 2 (<, «)s(3 y) t (DeftPrad 2 {t, n, y) V DeftexpFu 2
(t, n,y) V DeftrekFu 2 {t, n,y)')
Explanation: t is an w-termed symbol (pr’‘ or fu") which is de-
fined2.
D 76. Z 2 (f, n) = [UndPrad (f, «) V UndFu (f, «) V Deft 2 (/, n)]
Explanation: t is called an w-termed symbol2 when t is either a
pt" or an fu" and is either undefined or defined2.
D 77. Konstr 2 (r) = (Konstrl (r) • (x) r (t) x (y) x (m) t (n) Ing (3;)
[(AlnAR {x, r) . (^=zus [reihe (t), einkl (3^)]) • Z 2 {t, m) •
VR(3/,n))D(»i = «)])
D 78 : KonstrA 2 (jc), is analogous to D 52.
Explanation, D 77 : An construction! r is a construction! which
fulfils the following condition. In each expression occurring
in r, where di is an wi-termed symbol2 and an n-termed variable-
series, m is equal to w. Thus, in a construction2, every pt and every
fu has the correct number of arguments. — D 78 : The last expression
of a construction2 is called constructed2.
D 79. UndDeskr (/) = (3 w) < (UndPrad n) V UndFu (f, n) )
Explanation: t is an undefined descriptive symbol (namely pr or fu).
D 80. Undeft (t)= [(< = 4) V (# = 6) V (r= 10) V (r= 12) v (r= 14) v
(^=15) V (r = i8) V (r=2o) V (r = 2i) V (r = 22) V (r = 24) V
{t = 26) War (r) V UndDeskr (r)]
Explanation: t is an undefined '^'^symhol when t is either one of the
twelve undefined logical constants (see p. 55), or a variable, or an
undefined descriptive symbol.
D 81. DefKette (r) = (n) Ing (r) {x) gl (w, r) (t) x [(^ — {n = 0 ) •
[gl (n, r) = x\m InA (r, x)) D (KonstrA 2 (jc) • (3 ^ (Def 2
(*, s)) . [Undeft {t) V (g w) n (Def 2 [gl {m, r), t] )] . {/) Ing (x)
[(lDefrekFu2 (jc, /, t) 3 2 DefrekFu 2 [gl («', r), I, <]) .
( 2 DefrekFu 2 (x, I, t)o {2 m) n ([« = «'] . lDefrekFu 2
[gl(m,r), /,<]))])]
Explanation: r is called an ^^^definition-chain when the following
is true. Every SN^xpression occurring as a member of the chain r
is constructed2 and is a definition-sentence2. If t is a TNsymbol in
an expression which is a member of r, then either t is undefined,
or or some previous expression of r is a definition-sentence2 of U
72 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
If an expression of r is the first part of a regressive definition2, then
the expression which immediately follows is the second part of this
definition; if an expression is the second part of a regressive de-
finition2, then the immediately preceding expression is the first part
of this definition.
D82. DeftKette (/, r) = (3 x) r [DefKette (r) • [letzt {r)=x] •
Def 2 (^,/)]
D 83. Deft (0 = (3 0 pot (2, pot [2, pot (2, pot [2 pot (2, 0 ])])
[DeftKette (^,r)]
Explanation^ D 82 : t is defined by means of the definition-chain r.
— D 83 : A symbol t is called defined when there is a definition-chain
r by means of which t is defined.
D84. Bas (0 = (Undeft( 0 VDeft( 0 )
Explanation: A symbol t is called based either when it is undefined
or when it is defined (by means of a definition-chain).
D 85. Konstr (r) = [Konstr 2 (r) • (/) r (In AR (^, r) D Bas (t) )]
O 86 : KonstrA (at;), is analogous to D 52.
Explanation^ D 85 : A construction of expressions is a construction2
of which all the symbols are based symbols. — D 86 : An expression
is called constructed when it is the last expression of a construction.
D 87. ZA (oc) = (ZAl {x) • KonstrA (x))
D 88. Satz (x) = (Satzl (at;) • KonstrA (at;) )
Explanation: x is called a 3 (or an 8, respectively) when at; is both
a 31 (or an Si) and constructed. Thus the most important concepts
of the rules of formation Ore attained', in contradistinction to the pre-
viously defined auxiliary terms (3l, 3^, 31, S2), ‘ZA* and ‘Satz*
refer to based expressions only, and hence to the 3 (or to the S,
respectively) in the proper sense.
D 89. Def (of, t) = (Def 2 (x, t) • KonstrA (ac) )
D 90. Df (x) = (3 <) at; (Def (at;, t ) )
Explanation, D 89 : at; is a definition-sentence of t. (This definition
is analogous to D 87 and D 88.) — D 90 : of is a definition-sentence.
D91. DeskrZ (/)= (UndDeskr (t) V [Deft (t) • (r) — (Deft-
Kette (t, r) D (3 5) r [In AR (s, r) • UndDeskr (r)] )] )
D 92. DeskrA (at;) = ( 30 ^ (In A (/, at;) • DeskrZ (/) )
Explanation, D 91 : < is a descriptive symbol a^, either when t is
an undefined Cd or when t is defined and every definition-chain of
t contains an undefined ab (limit as in D 83). — D 92: at; is a de-
scriptive expression ^b when at; contains an Qb.
73
§22. FORMATION RULES! DEFINITIONS
D 93. LogZ {t) = (Bas (0 • DeskrZ (^))
D 94. LogA (x) = (/) X (inA (?, :ic) D LogZ (/) )
Explanation, D 93 : A logical symbol ai is based and not de-
scriptive. — D 94: jn? is a logical expression when all symbols of x
are logical symbols.
D 95. DeftZz (s) = (DeftZzl (5) • Bas {s ) )
D 96. Zz (s), D 97. Prad (s), and D 98. Fu (5), are analogous.
Explanation, D 95-98: Defined 33; 33; pr; fu. In contradistinc-
tion to the auxiliary terms which were defined at an earlier stage ,
the terms defined here refer to based TNsymbols only.
§ 23. Rules of Transformation*
The following definitions constitute the formalization of the
previously stated transformation-rules of Language I (§ 1 1 *and
§12). For this purpose substitution must first be defined (D 102);
D 99-101 introduce auxiliary terms for the definition of sub-
stitution.
D 99. I . stfrei ( 0 , s, x) = (Kn) Ing (jr) [Frei {s, x,n)m^ (3 tn) Ing (ap)
(Gr (m, n) • Frei (5, x, in))]
2. stfrei (^1 , at?) = (Kn) stfrei {k, s, x) (n = stfrei {k, s,x)')»
Frei (s, x, n) m ^ (3 fn) stfrei {k, s, x) [m = stfrei
{k, s, jc)] • Gr (m, n) • Frei (s, x, m) )]
D 100. anzfrei {s, x) = (Kn) Ing (a:) (stfrei (n, 5, ac) = 0 )
D loi. I . sb ( 0 , X, s,y) = x
2. sb(^',a:, j,jy) = ers (sb(^,ap,5, j), stfrei (^,^,a(;),jv)
Explanation, D 99-101 : Let s be ™3i. stfrei {k, s, x) is the position-
number of the (k + i)th 3i (counted from the end of the expression
x) which occurs freely in ac (0 in the case where there are not k-\- 1
free 3i in ac). anzfrei (s, x) is the number of the 3i which occur freely
* Key to the symbols :
stfrei : position-number of free 3
{Stellennummer des freien 3)
anzfrei: number of free 3 {Afi-
zahl freier 3)
sb, subst : substitution {Substitu-
tion)
GrS : primitive sentence {Grund-
satz)
AErs : expression- replacement
{Ausdrucksersetzung)
KV : no free variable (keine freie
Variable)
IJAblb: directly derivable (wn-
mittelbar ableitbar)
Abl : derivation {Ableitung)
Ablb : derivable {ableitbar)
Bew : proof {Beweis)
Bewb : demonstrable {beweisbar)
74 part II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
in X, sb(/f,jc,s,y) is that expression which results from the expres-
sion X when, starting with the last free 3i, the k last free 3i of x are
successively replaced by the expression y.
D 102. subst (jc, s,y) = sb (anzfrei {s, x), Xy s,y)
Explanation: If x is the SNexpression ^i*, ^» 3 r»
subst (x, 5, y) is the SNexpression (On substitution, see p. 22.)
D 103. GrSl (jc) = (3 y) jc (3 ar) jc [Satz (x) • (A; = imp (y, imp
[neg(3'),2]))]
Correspondingly D 104-113 : GrS 2 (jc) to GrSl 1 (j:) ; to give one
further example :
D io6. GrS 4 (jc) = (3 5) jc (3 y) x [Satz (:c) • (jc = ^ (zus [reihe 4
(6, i , I o, 4), einkl (3;)] , subst [y , i, reihe (4)] ))]
D 114. GrS (jc) = (GrSl (*) V GrS 2 (*) V ... V GrSl 1 (jc))
Explanaticnty D 1 03-1 13: ac is a primitive sentence of the first
kind; second kind; ...eleventh kind (PSI i-ii). — D 114: x is a
primitive sentence.
D 115. AErs {xi, Xi, Wj, »,) = {3 «) iCi (3 v) Xy [(ati = zus 3 («, Wj, ®)) .
(ac, = zu 83 («,ai2,®))]
Explanation: Expression-replacement: X2 results from Xx when
the partial expression is replaced by Wa* (In the case of the term
‘ ers * a symbol, whereas here an expression, is replaced.)
D 116. KV (y, Xy s) = ^ (3 n) Ing (x) (3 t) y (Fr (t, y) • Geb
(ty Xy n)m Frei (Sy Xy w))
Explanation: ‘ KV (y,ac,s) ’ means that no variable which is bound
in jc at a place of substitution for s occurs as a free variable in y.
(See p. 22.)
D 1 17. UAblbl (2:, ac) = (3 y) 2r (3 5) ap [ZA(y). (5r = subst(at:,^,y)).
KV(y,*,j)]
D 1 18- UAblb 2 (js, a;) = (3 %) sum (ac, ;?) (3 ^2) sum (at:, 2) (3 u) Wi
(3 v) ([ ([to, = imp (m, v)] . (tOj = dis [neg (m), w]) ) V ([to, =
kon (m, ®)] . [tOg = neg (dis [neg («), neg (»)]) ]) V ([to, = aq
(w, v)] • (fV2 = kon [imp (.m, v)y imp (Vy «)]))] • [AErs {x, z,
to,, to ) V AErs {x, z, to„ to,)])
D 119. UAblb 3 {z, x,y) =(x = imp(y, ar))
D 120. UAblb 4 (2:, Xy y) = {2 s) z [(* = subst [Zy Sy reihe (4)]) •
(y = imp (zy subst [Zy Sy reihe2 (r, 14)] ))]
D 121. UAblb {Zy Xy y) = (UAblbl {Zy x) V UAblb 2 {zy x) V UAblbS
{Zy af,y) V UAblb 4 (jsr, at:,y))
§23. TRANSFORMATION RULES 75
Explanation^ D 117: 2 is called directly-derivablel from x when
X is 9li and z has the form (according to RI i ; see § 12). —
D 118-120: ‘ directly-derivahle2 (or 3, 4, respectively)* in accord-
ance with RI 2, 3, 4. — D 12 1 : z is directly derivable from x or
from X and y.
D 122. Abl (r, p) = (3 ?) r (n) Ing (r) (*) r ([r = zus (/>, q)] . ~ [Ing
(r) = 0] . [ (-^ (n = 0) . [gl (n, r) = *] ) D (Satz (») . [Gr [n, Ing
(/.)] D (GrS (*) V Df (jc) V (3 *) n (3 /) n = n) . ~ (/= n) .
UAbIb[*,gl(A,r),gl(/,r)]])])])
Explanation: r is an ssN^erivation having the ssNseries of pre-
misses py if the following conditions hold : r is composed of p and q ;
every expression which is a member of r is a sentence; every ex-
pression which is a member of q is either a primitive sentence or a
definition-sentence, or is directly derivable from one or two pre-
vious sentences in r (see p. 29).
D 123. AblSatz (r, x,p) = (Abl (r,/)) • [letzt (r) = x])
Explanation: r is a derivation of the sentence x from the series of
premisses p,
D 124. Bew (r) = Abl (r, 0)
D 125. BewSatz (r, x) = (Bew (r) • [letzt (r) = «] )
Explanation, D 124: r is a proof when r is a derivation without
premisses. — D 1 25 : r is a proof of the sentence x.
Let ‘Ablb(jc,p)* mean: x is derivable from the series of pre-
misses p\ and ‘Bewb(:i;)’: x is demonstrable. These syntactical
concepts which refer to Language J cannot be defined in I. The
definitions are as follows :
Ablb (x,p) = (3 r) (AblSatz (r, x,p))
Bewb (:c) = (3 r) (BewSatz (r, x) )
For the formulation of these definitions, the unlimited opera-
tors, which do not occur in Language I, are required. The con-
cepts ‘derivable* and ‘demonstrable* are indefinite. In I only
definite concepts of dcrivability and dcmonstrability can be de-
fined ; for instance, such as refer to the derivation itself, or to the
proof itself, respectively (see D 123, D 125), or concepts like
‘ derivable from p by means of a derivation consisting of at most
n symbols*, or ‘demonstrable by means of a proof consisting of
at most n symbols*. If indefinite syntactical concepts are to be
76 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
defined as well, then an indefinite language must be taken as the
syntax-language — such as, for instance, our Language 11 .
For certain indefinite concepts, although they cannot be defined
in I, the universal sentence which states that they are predicable
for every single case can, however, be formulated in 1 . In the de-
finition of concepts like * not demonstrable * and ‘ not derivable * in
the indefinite language, a negated unlimited existential operator,
which can be replaced by a universal operator, occurs; and un-
limited universality can be expressed in I by means of a free
variable. ‘ ^ BewSatz (r, a) * means : “ Every r is not a proof of a
in other words: “a is not demonstrable**; ‘ ^AblSatz[r,b,reihe
(a)]* means: “Every r is not a derivation of b from a**, in other
words: “b is not derivable from a**.
§ 24. Descriptive Syntax
We have now completed our exposition of the pure syntax of
Language I ; this example makes it clear that pure syntax is nothing
other than a part of arithmetic. Descriptive syntax^ on the other
hand, uses descriptive symbols as well, and by so doing goes be-
yond the boundaries of arithmetic. For instance, a sentence of
descriptive syntax may state that at a particular place a linguistic
expression of such and such a form occurs. It has been pointed
out earlier (p, 54) that a possible method is to introduce a series
of undefined ptb as additional primitive symbols (for instance:
‘Var*, ‘Id*, ‘Prad*, and so on). But, as we have already an-
nounced (p. 54), we shall proceed differently. We shall take the
undefined fub *zei* as the only additional primitive symbol. (If
the sentences in which this symbol occurs are, in their turn, syn-
tactically treated, we shall co-ordinate to it the term-number
243 ( = 3^)*) 'T'he construction of descriptive syntax takes exactly
the same form as the construction of any other descriptive axiomatic
system A. First the syntax of the language S in which A is to be
formulated must be established. In this way the method of formu-
lating sentences and of deriving them from A is determined. For
some A (for instance, geometry and syntax) it is necessary that S
should contain an arithmetic.
The following basis of A will now be established in S : i . the
descriptive primitive symbols of A which are added to the primitive
77
§24- DESCRIPTIVE SYNTAX
symbols of S ; from these, according to the syntactical rules of S,
further symbols can be defined; 2. the axioms as additional primi-
tive sentences of S ; from these, with the help of the transformation -
rules of S, consequences can be derived (the so-called theorems of
A); 3. additional rules of inference \ in most cases, however, these
are not introduced. If we use undefined ptb as primitive symbols
of descriptive syntax, then a large number of axioms is necessary ;
by means of these it is stated, for instance, that unlike symbols may
not occur at the same place, and so on. Further, a number of
axioms in the form of unrestricted existential sentences is required,
in order to make it possible to derive even simple sentences about
derivability and demonstrability. If, on the other hand, we take
the fiib ‘ zei ’ as a primitive symbol, then no axioms of any kind are
necessary. That which in the other case is excluded by the .ne-
gative axioms is here already excluded by means of the syntactical
rules concerning functors (a particular fu can only have one value
for a particular place); the necessary existential sentences follow
from the arithmetic.
With the help of the primitive symbol ‘ zei \ we shall here give
the definition — a regressive one — of only one further symbol be-
longing to descriptive syntax. This is the fu^b ‘ ausdr * (Ausdruck) —
the most important term of descriptive syntax.
D 126. I . ausdr ( 0 , x) = pot [2, zei (x)]
2. ausdr (/e*, jc) = prod [ausdr (^, x)y pot (prim (A“), zei
[sum (*,*')])]
Explanation: ausdr (^,ac) is the expression (with i symbols)
which occurs at the positions x to x + k. Since the TNsymbol at the
position y is zei(y), therefore ausdr(^,x) = 2*®*^*^«
• prim(^l)^®*^* (see p. 56).
With the help of the functors ‘zei’ and ‘ausdr’, together with
that of the previously defined symbols of pure syntax (D 1-125),
we are now in a position to formulate sentences of the descriptive
syntax of I in I itself.
A. Examples of sentences about individual symbols (with the. help
of ‘zei’):
1 . “A symbol of negation occurs at the position a ” : ‘ zei (a) = 21 ’.
2. “Equal symbols occur at the positions a and b”: ‘zei(a) =
zei (b)
78 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
B. Examples of sentences about expressions (with the help of
‘ausdr*):
1. “In the series of positions a to a + b occurs a 3 ”-
*ZA(ausdr(b, a)) *.
2. “ a demonstrable sentence does not occur” : ‘ ^ BewSatz
(r,ausdr(b,a))* (with the free variable ‘r’, see p. 76).
§ 25. Arithmetical, Axiomatic and
Physical Syntax
Within the domain of descriptive syntax we can distinguish two
different theories : the axiomatic syntax which we have just been
discussing (with or without axioms) and physical syntax. The
latter is to the former as physical geometry is to axiomatic geo-
metry. Physical geometry results from axiomatic geometry by
means of t^ie establishment of the so-called correlative definitions
(cf. Reichenbach [Axiomatik]^ [Philosophie]). These definitions de-
termine to which of the physical concepts (either of physics or of
everyday language) the axiomatic primitive symbols are to be
equivalent in meaning. It is only by means of these definitions
that the axiomatic system is applicable to empirical sentences.
The following scheinatic survey is intended to exhibit more
clearly the character of the three kinds of syntax, by means of the
analogy with the three kinds of geometry. In addition, it is meant
to show the relation which subsists generally between arithmetic,
an axiomatic system, and the empirical application of the latter.
The three kinds of geometry. '
I. Arithmetical geometry,
A partial domain of arithmetic which
(in the usual method of arithmetiza-
tion, namely by means of co-ordinates)
is concerned with ordered triads of real
numbers, the linear equations occur-
ring between them, and the like.
The three kinds of ssmtaz.
I. Arithmetical {or pure) syntax,
A partial domain of arithmetic which
(in the method of arithmetization pre-
viously explained) is concerned with
certain products of certain powers of
prime numbers, the relations between
such products, and so on.
This partial domain is selected by means of certain purely arithmetical
definitions. The practical reason for framing precisely these definitions is given
by a certain model, namely, a system of physical structures for the theoretical
treatment of which these definitions are appropriate. This is the system
of physico-spatial relations which is the of physical linguistic structures — e.g.
subject of physical geometry, I IB. the sentences occurring on a sheet of
paper — ^which is the subject of physical
syntax, II B.
§25* ARITHMETICAL, AXIOMATIC AND PHYSICAL SYNTAX 79
II. Descriptive geometry.
(This designation is here not in-
tended in the usual sense, but in the
sense of the syntactical term ‘de-
scriptive ’.)
II. Descriptive syntax.
II A. Axiomatic geometry.
1 1 A. Axiomatic syntax.
Two different representational forms:
(u) Proper axiomatiza-
tion (compare § 18)
(‘ Axiomatized descrip-
tive syntax *).
(6) Arithmetization
(compare §§ 19, 24)
(‘ Arithmetized descrip-
tive syntax*).
A language with established logical primitive symbols, primitive sentences,
and rules of inference is presupposed for the axiomatic system.
Basis of the axiomatic system :
I. Axiomatic primitive symbols (descriptive primitive symbols which are added
to the primitive symbols of the language) :
“Point”, “straight
line”, “between”, and
so on.
‘Var*, ‘Nu*, ‘PrSd*,
‘ G 1 ’ (positions with equal
symbols), and so on.
‘ zei ’ as the only primi-
tive symbol. •
2. Axioms (descriptive primitive sentences which are added to the primitive
sentences of the language) :
For example, Hilbert’s
axioms.
Numerous axioms, for
instance: “a 3 is not a
pr”, “Gl(x,y)DGl(3;,
x)”, and so on.
No axioms.
Valid descriptive sentences of the axiomatic system :
I. Analytic sentences. For proofs of these the definitions belonging to the
axiomatic system may be used, but not the axioms themselves.
Examples. “ Every
point is a point”; “If
each of three straight
lines intersects the other
two at different points,
then the segments be-
tween the points of inter-
section form a triangle”
(this follows from the de-
finition of “ triangle”).
Examples. ‘Var(x)D Examples. ‘zei(») =
Var(Ap)*;‘Nu(x)DZz(x)’ zei(x)’; ‘[zei(jc)=4]D
Zz [zei (x)] *
(that is to say, “nu is a 33”; this follows from the
definition of ‘ Zz *) ;
‘ [Nu (x) • Str (xl)] D I ‘ ([zei (x) = 4] • [zei (x*)
ZA(x, i)’ I = i4])OZA[ausdr(i,x)]’
(that is to say, “nu* is a 3”; follows from the
definition of ‘ ZA * ; here ‘ ZA * is
a ptb). I a pti).
2. Synthetic sentences. These are the axioms themselves together with the
synthetic sentences which are proved with their assistance.
Example. “The sum
of the angles of a triangle
is equal to 2R.”
Example. ‘Nu(x)D'^
Ex(x)* (that is to say,
“an nu is not a ‘3 **’)•
None. Since there are
no axioms here, all valid
sentences are analytic.
II B. Physical geometry. | II B. Physical syntax.
By means of correlative definitions it is determined which symbols of the
physical language are to correspond to the primitive symbols (or to certain
defined symbols) of the axiomatic system.
8o PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
Examples. Examples. Examples.
1. “A physical seg- i. “*Nu(jc)* is to be i. “‘zei(jic)=4* is to
ment (for instance, the taken as true be taken as true
edge of a body) is said to ^hen and only when a written character having the
have the length i when figure of an upright ellipse (*0*) is to be found at the
It IS such and such a position it.”
number of tunes longer / v. • / x
than the wave-length of Nu(xi) is to be 2. zei(jf)— 4 is to
such and such a spectral true be taken as true
line of cadmium.” when and only when a character which has a suf-
2. “A physical seg- ficient resemblance in design to the character
ment is said to have the occurring at such and such a place (for instance of
length I when it is con- this book) is to be found at the positioii x.”
gruent with the segment
between the two marks
on the standard metre
measure in Paris.”
3. “Physical objects of
such and such a kind (for
instance, light-rays in a
vacuum or stretched
strings) are to be con-
sidered as straight seg-
ments.”
[Examples (1) are qualitative definitions \ here the term is defined by the state-
ment of the properties which an object must have in order to be comprehended
by the term. Examples (2) are ostensive dilutions ; here the term is defined by
the stipulation that the objects comprehended by the term must have a certain
relation (for instance, congruence or likeness) to a certain indicated object; in
linguistic formulation the ostension takes the form of a statement of the spatio-
temporal position. It is to be noted that, according to this, an ostensive definition
likewise defines a symbol by means of other symbols (and not by means of extra-
linguistic things).]
Valid descriptive sentences
I. Analytic sentences. These are either analytic sentences of the axiomatic
system, of which the axiomatic terms have acquired a physical sense by means of
the correlative definitions (Examples {a) ; compare the examples of analytic sen-
tences under II A), or on the other hand (Examples (6)) sentences which are
translated from such sentences, by means of the correlative definitions, into the
non-axiomatic terminology (that is to say, into a terminology which does not
belong to the axiomatic system, but to the general language).
Examples. Examples. Examples.
(a) “If each of three (a) “A zero symbol (a) “A (physical) ob-
^hysical) straight lines (physical character in ject which possesses the
intersects the other two ink) is a numeral.” term-number 4 (that is,
at different points, then a certain physical pro-
the (physical) segments perty) is a numeral.”
between the points of in-
tersection form a (physi-
cal) triangle.”
(b) “If each of three (6) “A (physical) character having the design of
light-rays in a vacuum an upright ellipse is a numeral.”
intersects the other two
§25- ARITHMETICAL, AXIOMATIC AND PHYSICAL SYNTAX gl
at different points, then
the segments of rays be-
tween the points of inter-
section form a triangle.**
2. Valid laws. I’hese are either indefinite synthetic sentences of the axiomatic
system, which in this case have a physical meaning (Examples i (a), 2 (a)), or
translations of such into non-axiomatic terminology (Examples i (6), z (6)).
Examples. Examples. None, because there
I (fl). “ T ATO (physical) i (a). “ If a (physical) are no axioms,
straight lines intersect zero symbol occurs at a
one another at one point place, then no existential
at most.*’ symbol occurs there.**
1 {h). “ Two light-rays i (6). “ If a character
in a vacuum intersect one in ink having the figure
another at one point at of an upright ellipse oc-
most.** curs at a place, then no
2 (fl). “ The sum of the character consisting of
angles of a (physical) tri- one vertical and three
angle is zR.” horizontal strokes occurs ^
2 {b). “ The sum of the at that place.**
angles between three
light-rays in a vacuum
which intersect one an-
other is zR.”
The question of the validity of a particular axiomatic system having certain
correlative definitions is the question of the validity of the laws which result
from the translation of the axioms into the language of science (of physics)
(Example i (6)).
Here arises, for in- Here the question of
stance, the important validity is a critical one
question of validity in in relation to the existen-
relation to Euclidean or tial axioms, and particu-
to one particular non- larly to the axioms of
Euclidean geometry. infinity (for instance,
“ there are infinitely
many variables **).
3. Empirical sentences. Hereby are to be understood definite synthetic sen-
tences which state the empirical (namely, the geometrical or graphical) properties
of certain physical objects, whether demonstrable by means of the axioms or not.
The sentences may either employ the non-axiomatic terminology (Examples
I {a)y 2 {a)) or be translated into the axiomatic (geometrical or syntactical)
terminology (Examples i (6), 2 {b)).
Examples. | Examples.
I (a). “A symbol consisting of two horizontal
strokes occurs at the place c in this book.”
1 (6). “A symbol of identity occurs at the place c
in this book**; in the symbols of our system:
‘Id(c)*. I ‘zei(c) = is’.
2 (a). “A series of figures of such and such a form
occurs in the places ranging from a to b in this
book.**
Examples.
I (a). “ This object A
is a light-ray in a vacu-
um.”
I (6). “A forms a
straight segment.”
2(a). ” rhese three
objects A, B, C are light-
rays in a vacuum each
one of which intersects
Here there is no ques-
tion of validity at all.
(On the dispensability of
an axiom of infinity for
arithmetic, see p. 97.)
SL
7
82 PART II. FORMAL CONSTRUCTION OF THE SYNTAX OF I
the other two at different
points."
2(b), “The physical
objects A, B, C together
form a triangle."
2(b), “A primitive sentence of Language I
occurs...."
The following sentences are of a like kind :
3. “The sentence ^docendo discimus* occurs in
that book."
4. “ It is maintained in that book that one learns
by teaching."
5. “In such and such a treatise, the sentences
occurring at places so and so contradict one another."
6. “The word-series at such and such a place is
meaningless (that is to say, is not a sentence of such
and such a language)."
7. “An empirically false sentence occurs at such
and such a place." (Cf. ‘ P-contravalid p. 185.)
The sentences of the whole history of language
and literature belong here, especially those of the
history of science, including mathematics and meta-
physics. Among them are both sentences which
merely cite something (Examples 2 (o), 3) and sen-
tences (Examples 2 (6), 4 to 7) which presuppose the
syntax of the language in question and sometimes
also certain synthetic premisses, particularly such as
criticize formulations and theses on the basis of
logical analysis (Examples 5, 6) or of experience
(Example 7).
PART m
THE INDEFINITE LANGUAGE H
A. RULES OF FORMATION FOR LANGUAGE II
§ 26. The Symbolic Apparatus of Language II
Language I, with which we have been concerned up to the
present, contains only definite concepts ; in the domain of mathe-
matics it contains only the arithmetic of the natural numbers, and
that only to an extent which corresponds approximately tP a
finitist or intuitionist standpoint. Language II includes Language
1 as a sub-language ; all the symbols of I are likewise symbols of
II, and all the sentences of I are likewise sentences bf II. But
Language II is far richer in modes of expression than Language I. It
also contains indefinite concepts; it includes the whole of classical
mathematics (functions with real and complex arguments ; limiting
values; the infinitesimal calculus; the theory of aggregates); and
in it, in addition, the sentences of physics may be formulated.
We shall first state the symbols and the most important ex-
pressions which occur in Language II. The exact rules of forma-
tion for 3 S will be given later (§ 28). The Gothic symbols
used in the syntax of Language I will also be used here, together
with some additional ones.
In Language II, in addition to the limited operators of Lan-
guage I, we have also unlimited operators of the forms (3), (3 3), and
(K3). Example: ‘ (3 x) (Prim {x)y ; see § 6. ]
In Language II, fu and pr of new syntactical kinds occur, and
these are divided into levels and types (§ 27). In the sentence
fu(2Ii)=9l2, we shall, as hitherto, call 9 li the argument-expression;
and, further, we shall call 2I2 the value-expression. In II there are
fu in which not only consists of several terms — ^the so-Called
arguments — ^but 3I2 also consists of several terms — ^the so-called
value-terms [33, and 35 in fu(3i,32) = 39.3«i3*. fo*"
Stance]. There are not only the predicates pr but also predicate-
expressions (of the different types) which may consist of several
84 PART III. THE INDEFINITE LANGUAGE II
symbols, but are used syntactically just like the pr. Further, there
are functor-eocpressions gfu (of the different types) which are used
syntactically like the fu (examples will be given later). Just as a
one-symbol expression 53 is a 3> so a pt is a and an fu an
There are pr (and other ^r) of which the arguments are not 3 hut
either or gfu (of one type or another); further, there are fu
(and other gu) of which the arguments and value-terms are not 3
but are either or gfU (of one type or another). Thus, an argu-
ment-expression or a value-expression (syntactical designation,
‘ Slrg *) consists of one or more expressions of the forms 3» or
^U, separated from one another by commas.
In Language II, there are variables of different kinds : not only
numerical variables 3 (‘ u ‘ u ;ar *), but also predicate-variables
and functor-variables f (7*, ‘A’).
[Just as we assign the 3 to the 33, so we assign the p to the pr and
the f to the fu.] The variables p and f (of all types) also occur in
unlimited operators: (p); (3p); (f); Of).
In Language II, the symbol of identity ‘ * is not only used
between 3 and between S (here also when used between S it is
usually written ‘ = ’) but also between and between 3fu.
\Examples (for the simplest type): = P2’ is equivalent in mean-
ing to ‘(jc) (Pi(jc) = P2(a[;))’; ‘fui = fu2* is equivalent in meaning to
‘(«) (fui(:c) = fu2(jip))'.] We shall designate thea:«^o-^^fl/w « ‘0 = 0 *
by ‘ 91 *.
In Language II, ' sentential symbols [Satzzeichen] fa also
occur; these are in part sentential constants y that is to say symbols
which are used as abbreviations for certain sentences, and in part
sentential variables f ‘ The f also occur in operators
of the form (f) and (3f). We use ‘n* as the common designation
for the variables of the four kinds which we have mentioned,
namely, 3, p, f, f ; all the remaining symbols are constants if).
§ 27. The Classification of Types
Every ^r, and hence every pr and every p, belong to a certain
type. Further, we assign a type to all the 3 — namely, the type 0 .
A particular ^r can only have arguments of certain types, and an
gu can only have arguments and value-terms of certain types.
In order that and maybe
§ 27. THE CLASSIFICATION OF TYPES 85
sentences, it is necessary that and should belong to the same
type, and furthermore that % and should belong to the same
type (which may, however, be of another kind than that to which
belongs), and so on. In order that 5 ui( 3 li, • • . ...
?lm+n and 8^1 may be sentences,
91 ,. and 91 ',- (i= i to m + «) must be of the same type. The type of
a is determined by the types of the arguments (in which
number and order must be taken into account) ; the type of an
3fu is determined by the types both of the arguments and of the
value-terms.
The t3rpc of an expression is determined in accordance with the
followii^g rules. Every 3 (and hence also every 33) belongs to the
type 0 . If the n terms of an 9 Irg have the types /j, ^2, ... (in this
order), we assign the type ^1,^2, ... tn to the 9 lrg. [The symbols
with suffix are not themselves syntactical type-designations,
but are syntactical variables of such.] If 9 Irgi belongs to the type
ti in the sentence ?Jri( 9 Irgi), then we assign the fype (f^) to
If 9 lrgi belongs to the type ti and 9Irg2 to the type ^2 in the
sentence 8fUi(9Irgi)=9lrg2, then we assign the type (^11/2) to
3fUi and the type /2 to the expression 3fUi(?Irgi).
Examples, i. ‘ Gr (5, 3) * is a sentence ; ‘ 5 * belongs to the type 0,
‘ 5, 3 * to the type 0, 0 ; hence the pr * Gr * belongs to the type (0, 0).
— 2. ‘sum (2, 3) = ac* is a sentence; therefore the fu‘sum’ belongs to
the type (0, 0 : 0). — 3. Let ‘ M ’ be a pr of which the arguments are
not 3 but a pr and an fu of the types just mentioned, so that, for
instance, ‘M(Gr,sum)’ is a sentence. Then ‘M* belongs to the
type ((0,0), (0,0:0)).
The level-number of an expression is also determined by its
type, in accordance with the following rules. We assign the level-
number 0 to the 3- The level-number of an 9lrg is equal to the
greatest level-number of its terms. The level-number of a is
greater by i than that of the argument-expression belonging to it.
The level-number of an JJu is gi eater by i than the greatest of
the two 9 Irg belonging to it. In accordance with our previous rules,
every>^ type-designation, apart from commas and colons, consists
of zeros and brackets. The level-number is easily obtainable from
a designation of this kind; it is the largest number of pairs of
brackets in which a zero of the type-designation is included. To
the Gothic symbols, ‘ ipr * and so on, we append (as before), where
86
PART III. THE INDEFINITE LANGUAGE II
necessary, indices in the right-hand upper comer to designate the
number of argument-terms, and further, where necessary, indices
in the left-hand upper comer to designate the level-number.
The classification of types outlined above is, in its essential points,
the so-called simple classification of types proposed by Ramsey. But
it is here completed by being extended not only to pr but to fu,
and Sru as well ; and further, by the introduction of type-designa-
tions. In Russell’s so-called ** branched” classification of types, the
pt are further subdivided so that not only the t3rpe of the arguments
of a pi is taken into consideration but also the form of its definition.
Further, in his system the sentences also are subdivided into types,
whereas in our Language II any sentence can be substituted for an
f. In order to avoid certain difficulties which arise in the application
of his branched subdivision, Russell formulated the Axiom of Re~
ducibUity. But this axiom is rendered unnecessary by restricting
ourselves to the simple classification of types.
Examples, i. ‘ Gr * belongs to the type (0, 0) (see above), and thus
has the level-number i ; ‘Gr’ is thus a ^pr^, or, in words, a two-
termed predicate of the first level. — 2. Since every 3 belongs to the
type 0 and has the level-number 0, all the pr in Language I are 'pr ;
the types which occur are as follows : (0) ; (0, 0) ; (0, 0, 0), and so on.
All the fu of Language I are 'fu and of the types : (0 ; 0) ; (0, 0:0);
(0, 0, 0 ; 0), etc. — 3. In ihe example (3) above, ‘ M ’ is a ^pr^. — ^4. If,
in any connection, sentences of the form (jj) (pti (51) D pig (5i))
occur frequently, then, for the purpose of abbreviation, it is ex-
pedient to introduce the pr ‘Sub’ (“...is a partial property or a
sub-class of...”); the definition is as follows: ‘Sub(F, C^ = (x)
[F(x)dG(x)]*. Since ‘F’ and ‘G’ in this case are 'pr' of the type
(0), ‘Sub’ is a ®pr*,of the type ((0),(0)). — 5. Let ‘(^c) [(Pi(x) v
Pj (jc)) = P3 (x)] ’ be demonstrable. In accordance with the termino-
logy of the theory of aggregates or classes, we may here designate
P3 (the property or class) as the sum of Pj and Pj. For the purpose
of abbreviation, we propose to introduce the symbol ‘ sm ’ in such
a way that the expression ‘ sm (Pi, P2) ’ means the sum of Pi and Pj,
and hence, in the case given, is equivalent in meaning to ‘Ps’.
‘sm(Pi,P2)’ is accordingly a ^r belonging to the same type as
‘Ps’, i.e. to (0). The above-mentioned demonstrable sentence may
now be formulated more shortly, as follows: ‘ sm (Pi, Pj) = Ps
‘ sm * is an f u ; each of the two arguments as well as the value-term
belongs to the type (0) ; hence ‘ sm ’ is a *fu* of the type ((0), (0) : (0)).
The definition of ‘ sm ’ is as follows : ‘ sm (F, G) (x) = (F(x) v G (x)) ’.
Here the 5pr ‘ sm (F, G) * is used syntactically in the same way as a
pr of the same type (0). — 6. Let ‘F’ be a 'p of the type (0) and ‘ Cl’
a ^pr of the type ((0)) (in another mode of speech : a class of classes ;
see § 37), so that ‘ Cl (F) ’ is a sentence. * clsm (Cl) ’ represents the
class-sum of Cl ; by this is meant that property (or class) which is
applicable to all those numbers, and only to those, which have at
§27* THE CLASSIFICATION OF TYPES 87
least one property having the second-level property Cl. Let us
take * M' as a of the type ((0)); the definition will then be as
foUows: ‘clsm(M)(Jc) = ( 3 F)(M(F)*^M)’. ‘clsm(M)’ is a
of the type (0) ; hence ‘ clsm * is a ®fu of the type (((0)) : (0)). — 7. Let
*scn(F, G)' mean: the smallest common number of the two pro-
perties F and G ; and let it mean 0 for the case in which no such
number exists. The definition is as follows: ‘ sen (F, G) = (Kx)
(F(x) •G(x)) \ Each of the two arguments of ‘ sen * belongs to the
type (0). The value-expression of ‘ sen * (the right-hand side of the
equation) is a 3 therefore belongs to the type 0. Thus * sen ’
is a *fu of the type ((0), (0) : 0), and ‘ sen (F, G) * is likewise a 3 * —
Further examples will be given in § 37.
§ 28. Formation Rules for Numerical
Expressions and Sentences
On the basis of the foregoing explanations, which were bound
up with material interpretations, the rules of formation for Lan-
guage II may now be laid down formally in the following manner.
(Compare the analogous rules for Language 1 , § 9.)
We assume the previously given definitions of the following
concepts: ‘bound* and ‘free variables* (now with reference to all
0, namely 3, p, f, and [); ‘open* and ‘closed* (p. 21); ‘definite*
and ‘indefinite* (p. 45); ‘descriptive* and ‘logical* (p. 25);
‘33* and ‘St* (p. 26).
An expression belongs to the type 0 — in which case it is called
a numerical expression (3) — ^when and only when it has one of
the following forms: 1.33; 2. 3'; 3. (K3i)3i(S), or (K3i)(S),
where 3^ does not occur freely in 3i; 4* where Slj belongs
to any type and belongs to the type (^i :0), and is therefore
an gu.
In general, the following is true: if belongs to the type
and typ® (^1 • h ) — which case '2I2 is called a functor-
expression (gu) — ^then 312(^1) belongs to the type (but not only
in this case). The formation rule (+) which has already been given
for 3 is ^ special case of this. An expression of a type of the
form (ti), where is any type whatsoever, is called a predicate-
expression (^r).
Regressive rules for *n-termed argument-expression^ (or ‘value-
expression*) (3lrg”) are as follows: an 3 Irg^ has one of the forms
3> or gu. An 9 lrg"‘*‘^ has the form 3 trg”, 3 lrg^; if 3 Irgi and
88
PART III. THE INDEFINITE LANGUAGE II
^^2 belong to the types ti and ^2* respectively, then
^g2 belongs to the type
An expression is called a sentence (S) when and only when it
has one of the following forms: i. fa; 2. where and ^
are either 3, ?}r, or Qfu of the same type ; 3 . (S) or (S) ocrin (S) ;
4- (3i)3i (<») or (33i)3i (<3)» where 3i does not occur freely in
S- (») (S) or (3 0) (S); 6. ?ls(3Ii), where 3Ii belongs to any
type whatsoever and 5I2 belongs to the type (^i) (and is accord-
ingly a ^r).
Qi is called an atomic sentence when 6^ has any one of the
forms 91, pri(9Ii), or fUi(9l2)=9Is, where pti is an undefined
^ptb and fUi is an undefined ^fUb, and 9Ii, 9I2, and ^ are argument-
expressions of which all the terms are St. Si is called a molecular
sentence when Si is either an atomic sentence itself, or is formed
from one or more such by means of symbols of negation and
junction (and brackets).
Some of the syntactical definitions become simpler if we do not
consider the whole of Language II, but instead only certain con-
centric language-regions IIi, II2, ..., which form an infinite series.
As regards the apparatus of symbols, sentences, and derivations,
every region is contained in all the successive regions, and I is
contained in IIi. In a certain sense, Language II represents the
sum of all these regions. The subdivision into regions takes place
in the following way. Not counting pr and fu, all the symbols
already occur in IIi,' and thus in every region. Operators with f
occur for the first time in II2. In II^, ^pr and occur both as
constants and as free variables, but not as bound variables.
Further, in a region 11^ (^ = 2,3, ...) pr and fu occur as constants
and as free variables up to the level n, but as bound variables
only up to the level n — i. [The line of demarcation between II^
and the further regions corresponds approximately to that be-
tween Hilbert’s elementary and higher calculus of functions.]
§ 29. Formation Rules for Definitions
In Language II we shall admit only explicit definitions.^ This
involves no restriction, since, by the use of unlimited operators,
• {Note^ i93S0 I would now prefer to admit regressive definitions
of ^fu in II as in I. By that means the term 'definite’ (§ 43) would
§29. FORMATION RULES FOR DEFINITIONS 89
every regressive definition can be replaced by an explicit defini-
tion. Let fu^ be defined by means of a regressive definition which
is composed of and S2. From these sentences we construct
03 and ©4, by replacing fui by throughout, and then we define
fu^ by means of the following explicit definition (on ‘ ( ) * see p. 94 ;
here it is used only in relation to the 3) :
fU 2 (3l> • • • 3in) == (®^3n) (3 fl) [l ) (®8 • ^4) • == fl (3l» • • • 3m))]-
Then fUi=fU2 is demonstrable, and fu2 is thus equivalent in
meaning to fUi- Hence, the regressive definition can be replaced
by this explicit definition.
Primitive symbols in Language 11 : i . Twelve logical constants,
namely nu and the eleven individual symbols (as in Language 1 ,
see pp. i6 and 23) ; 2. all 0 ; 3. ptb and fUb, when and as required, of
any type. ['V* and '•* could also be introduced as defined sym-
bols, but we place them amongst the primitive symbols and state
their definitions as primitive sentences so as to be able to formu-
late the remaining primitive sentences more simply.*]
Formation rules for definitions. Every definition is a sentence of
the form is called the definiendum and ^ the definiens.
The symbol which is to be defined (a 33, pr, fu, ucrfn, or fa) only
occurs in ; beyond this, the only symbols which may occur in
9Ii are unequal variables as arguments, commas, and brackets.
No 0 which does not occur in occurs freely in Thus, a
defined fa is always an abbreviation for a closed sentence. [For
examples of definitions, see §§ 27 and 37.]
Since all definitions are explicit, it is in general possible to
eliminate a defined symbol Oi occurring in a sentence ©^ ; with the
following qualification : when Oi is a pr or an fu, the elimination
cannot be carried out from ©i, as it stands, if Oi occurs at least once
in ©1 without an ?lrg following it (that is, either as an argument
or as a value-term, or together with ‘ = ’). In order to dispose of
this difficulty, we can transform ©^ into ©g in the following
manner. We construct ©2 from by replacing Ui at all places at
have the intended wide extent. Dr. Tarski has pointed out to me
that the exclusion of regressive definitions would make even ‘ sum \
‘ prod *, and almost all arithmetical terms indefinite. — In the case of
the elimination required in RRi (§ 346) a regressive definition
would then, where necessary, have to be transformed, in the way
indicated in this, section, into an explicit definition.
90
PART III. THE INDEFINITE LANGUAGE II
which it occurs in Si without argument by a variable of the
same type (a p or an f), which does not otherwise occur in Si*
S3 is then constructed in the form:
(Ol)(»l) ••• = Vfn))^^-
Example, Let ‘Pg* be defined by means of Pg (jc) = (Pi (ac) •
cannot be immediately eliminated in *M(Pa)’ (6i)*
We transform Si into S3 thus: ‘(jc) (F(jtf) = P3(jc))DM*(iO*; the
elimination is then possible: ‘(x)(F(jc) = [Pi(jc)*P3(x)])3M(F)’-
B. RULES OF TRANSFORMATION
FOR LANGUAGE II
§ 30. The Primitive Sentences of Language II
To the rat^e of values of a variable 3, p, or f belong those ex-
pressions which are of the same type as the variable (thus to the
range of values of a 3 belong the 3 )* The S belong to the range of
values of an f.
Simple substitution. j ’ is a syntactical description of the
expression which results from ^ when Pi is replaced by ^1 at all
places at which it occurs freely in Here ^1 must be an ex-
pression from the range of values of Di which contains no free
variable that is bound at one of the places of substitution in
Substitution with arguments. is a syntactical
description of the expression constructed in the following
manner. Pi(^tgi) is a sentence; the terms of ^gi are unequal
variables, e.g. 0 i,02>"*^a;- is not necessary that these should
occur in 0^; on the other hand, free variables which do not occur
in ^tgi may occur in 0^, but these may only be such as are not
bound at the substitution-places in (that is to say, at those
places at which Px occurs freely in Px may not occur at any
substitution-place in without being followed by an argument-
expression. [An occurrence of this kind can, under certain cir-
cumstances, be obviated in the way described in § 29.] Unequal
^g may come after Px at the various substitution-places. Let
the argument-expression follow Px at a certain sub-
stitution-place. Then af this place Px (SIj, . . . 31 ^) is to be replaced by
§30. PRIMITIVE SENTENCES OF II 9 1
(^) ^ obtained when a replacement of this
kind in is carried out at all substitution-places.
Example. Let be ‘W (F(ac.3)) vF(0,2) v( 3 F) (M(F))’.
The substitution where ‘fu* is an fu, is to be carried
out. *F*v& only free in at the first and second occurrence ; there-
fore only these are substitution-places. Thus the fact that ‘F’ is
without ^rg at its third and fourth occurrence does not matter.
01 is *u = fu(jc)*. At the first substitution-place we must replace
‘F(jic, 3)* by ^bat is by'Si itself. Then, at the second
substitution-place, we must replace ‘F(0,j8f)* by which
is ‘ii = fu(0)*. The result of the substitution is the following:
‘(jc)(tt = fu(jc))v(M = fu(0))v(3F)(M(F))». The fact that the
variable which is bound at the first substitution-place, occurs
freely in the substitute expression here given, does not matter ; only
the “ surplus ” variable ‘ u * must not be bound at any of the substitu-
tion-places in
Primitive sentences of Language II. Since in Language II we
have the variables f and p at our disposal, we are able in many
cases to state a primitive sentence itself instead of a schema of
primitive sentences. PSII 1-3 and 7-14 correspond to the
schemata PSI i-ii of Language I (§ ii), PSII 10 and ii being
extended to the new kinds of variables.
(a) Primitive sentences of the sentential calculus.
PSIIi. pD(^pDq)
PSn 2 . {'^P 0 p)Dp
PSns. Ip 0 q) 0 (iq 0 r) 0 {p 0 r))
psn4.* ip^q)={^p:>q)
psn 5 . (/>«^)= V
Psn 6 . ((pOq).{q:>p))oip=q)
(b) Primitive sentences of the limited sentential operators.
PSOt. (*)0(F(*)) = F(0)
PSU8. (F(*))= [(*)>' (F(jc)).F(y)]
PSn9. (3*)>'(F(*))=~(*)>'(~F(*))
• {Note, 1935.) In the German original, GII4 (our PSII4) runs:
{pOq) = {'^pyq). For the reason for the change see the footnote on
P* 32 .
92
PART III. THE INDEFINITE LANGUAGE II
(c) Primitive sentences of identity,
PSnio. Every sentence of the form
PSn II. Every sentence of the form D
{d) Primitive sentences of arithmetic.
PSni2. ^( 0 = 3 ci)
PSni3. {x^=y^)D(x=y)
(e) Primitive sentences of the K-operators,
PSn 14. G ((K*)>- [f (*)])= [(~(3 x)y [F (*)] . G (0)) V (3 x)y
(Fix ) . (*) Jt [~ iz=x)o ~Fiz)] . G (*))]
PSn 15. G ((Kx) [F (*)])= [( ~ (3 X) [F (X)] . G (0)) V (3 x)
(F(x).(x)x[~(x=x)D ~F(x)].G(x))]
if) Primitive sentences of the unlimited sentential operators.
PSn 16. Every sentence of the form (di) (Si)d Si j
PSn 17. Every sentence of the form (pj) j
PSn 18. Every sentence of the form (3 oO (Sj) = ~ (t>i)
PSn 19. Every sentence of the form (Dj) (f^ V Sx)D[fiV(o0(<5i)].
where Vi is not
{g) Primitive sentence of complete induction,
psnao. [F(0).(x).(F(x)dF(x>))]3(x) (F(x))
(A) Primitive sentence of selection.
PSn 21. Every sentence of the form ((pj) [Pi (P2) 0 (3 Di) [P2 (Oi)] ] •
(P») (Ps) [(Pi (P*) • Pi (Ps) • (3 »i) [Pi (»i) • Pi K)]) 3
(Pi=P3)]) 3(3 PiXPi) (Pi (Pi) 0 [(3 Di) [Ps (»i) . P4(0i)] .
,{»l) (Ol) ([Pi (»l) • Pi (»l) • Pi (»i) • P4(»l)] 3 (Ol=l>!))]).
where x>i (and thus ^2 also) is either a p or an f .
(t) Primitive sentences of extensionaUty.
PSn 22. Every sentence of the form
(Ol) (Pl(0l) = Pi(0i))3(Pi = P2)
PSn 23. Every sentence of the form
(»l)(l>i)-(0n)(fl(0l. •••l>n) = fl(Dl. •••0«))3(fi = fi)
The variables which are named in the schemata may belong to
§ 30. PRIMITIVE SENTENCES OF II 93
any type whatsoever; the stipulation that the whole expression
must be a sentence is sufficient to secure the correct relationship
to one another of the types of the different variables. [For in-
stance, if, in PSII 21, Di belongs to the type (of any kind except
0), then it follows that Pg, Pg, and P4 must belong to the type (fj)
and Pi to the type ((^i))-] — PSII 4-6 are substitutes for definitions
of the junction-symbols ‘ V *, ‘ \ and ‘ = * (between S) ; they
correspond to RIza-t. PSII 6 need only be put down as an
implication; the converse implication follows with the help of
PSII II. — PSII 16 and 17 are the most important rules for the
unlimited universal operator \ by means of these schemata simple
substitution and substitution with arguments respectively are
rendered possible. — PSII 18 replaces an explicit definition of the
unlimited existential operator. — PSII 19 makes possible the so-
called shifting of the universal operator. — PSII 20 is the Principle
of Complete Induction ; in Language I it was formulated (RI 4)
as a rule of inference, but here, with the help of tho unlimited
operator, it can be formulated as a primitive sentence. — PSII 21
is Zermelo’s Principle of Selection (corresponding to Russell’s
Multiplicative Axiom) in a more generalized form (applied to
types of any kind whatsoever); it means: “If M is a class (of the
third or higher levels), and the classes which are elements of M
are not empty and are mutually exclusive, then there exists at least
one selective class, //, of M — ^that is to say, a class H which has
exactly one element in common with every class which is an
element of M.” If this sentence is applied to numbers as elements,
then it is demonstrable without PSII 21. (In such a case it is pos-
sible, for instance, to construct the selective class by taking the
smallest number out of every class which is an element of M.)
Therefore in PSII 21 it is established that Uj and Ug are not 3, but
are either p or f. — The formulation of PSII 22 (in conjunction
with PSII ii) effects the result that two pr which are co-extensive
are everywhere interchangeable, and therefore synonymous. Thus
all sentences of Language II are extensional with respect to
(see § 66). PSII 23 effects a corresponding result for the JJu- It
should be noted that an equation of the form 3i = 32> or
or gui = {Juz does not mean that the two terms of the equations
are equivalent in meaning. The two expressions are equivalent in
meaning (synonymous) when and only when the equation is analytic.
94
PART III. THE INDEFINITE LANGUAGE II
§31. The Rules of Inference of Language II
The rules of inference of Language II are very simple:
Rn 1. Rule of implication. Qs is called directly derivable from
and St when S2 has the form Si 3 S3.
lUla. Rule of the universal operator. Ss is called directly de-
rivable from Si when S3 has the form (0) (®i).
Only RI 3 ( = RII i) of the four rules of inference of Language I
(§ 12) is here retained. RI i is replaced by PSII 16 and 17 and
RII 2: from Si, according to RII 2, is derivable (Oi) (Si) or
(Pi) PSII 16 or 17 and RII i, are de-
rivable respectively. RI 2 is replaced by
PSII 4-6; RI 4 by PSII 20.
In the construction of a language, it is frequently a matter of
choice whether to give a certain rule the form of a Primitive
Sentence or that of a Rule of Inference. If it is possible without
too much complication, the first form is the one usually chosen.
\Example: In Language I the principle of complete induction can
only be formulated as a rule of inference; in Language II it may
be either a primitive sentence or a rule of inference. We have
chosen the former. Further examples emerge from a comparison
with other systems, see § 33.] It is, however, incorrect to hold
that there is a difference in principle, namely that for the
establishment of a rule the language of syntax (usually a word-
language) is necessary, while for the establishment of a primitive
sentence it is not. Actually, the latter must also be formulated in
the language of syntax, namely by means of the stipulation is
a primitive sentence” (or ”... is directly derivable from the null-
class”, compare. p. 171).
The ttnoA* derivation\ 'derivable', 'proof', 'demonstrable' have
the same definition here as they have in the syntax of Language I
(p. 29). If 0i,02, ... On variables of in the order of
their appearance, then '( ) (Si)’ will designate the closed sentence
(0i)(^>2)** (0n)(Si); if Si is closed, then ()(Si) is Si itself.
Si is called refutable when ^()(Si) is demonstrable. Si is
called resoluble when Si is either demonstrable or refutable;
otherwise it is called irresoluble. The terms ‘analytic’, ‘conse-
quence’, and so on wilt be defined later (§§ 34^,/).
§ 32. DERIVATIONS AND PROOFS IN II 95
§ 32. Derivations and Proofs in Language II
We will now give some simple theorems about demonstrability
and derivability in Language II. The proof and derivation sche-
mata are shortened.
Theorem 32,1. Every sentence having one of the following forms
is demonstrable in Language II :
{a) Si(5f)3(3t»x)(60-
Proof schema. PSII 16
(0
(I)
(*)
(2), Sentential Calculus (transposition)
(3)
(3). PSII 18
(4)
(^) M (®i)3(3 t>i) (®i)* From PSII 16, Theorem i a.
(0 (33i) (Bi= 3 i)- From PSII 10, RII 2, Theorem i b.
Theorem 32.2. In Language II is derivable:
{a) from Si 0621 where
x>i does not occur freely in
Si:
SiD(di) (S,).
Schema of derivation. Premiss: Si 002* 0i does not occur
freely in Si;
(I)
(i), Sentential Calculus
^SiV 02
(2)
(2), RII 2
(»i)(~< 5 iVS 0
(3)
(3). PSII 19
(4)
(4), Sentential Calculus
(5)
(A) from 01302, where
0i does not occur freely in
S,:
(30i)(<Si)3S*.
Schema of derivation. Premiss: Si 3 02, ©i does not occur
freely in Sj;
(I)
(i), Sentential Calculus
1
>
(2)
(2), RII 2
(i)i)(S*v~Sx)
(3)
96
PART III. THE INDEFINITE LANGUAGE II
(3) , PSII19 S.v(d,)(~< 5 ,) (4)
(4) , Sentential Calculus ~(»i) (S)
(5) , PSII18 ( 3 »i)(®i) 3 S* (6)
(c) from SiD(o)(S*): SiDS2.
Schema of derivation. Premiss; SiD(t>i)(®i) (0
PSII16 (*)
(i), (2), Sentential Calculus SiDS* (3)
Theorem 32.3. In Language II are mutually derivable:
(a) Si and (o)(Si); hence also Si and ()(( 3 i). By RII2
and PSII 16.
(*) (i)i)(t)2)(Si)and(D^(Oi)(Si).
Schema of derivation. Premiss: (Oi)(d*)(Si) (i)
1(1), twice PSII 16 Si (2)
(2), twice RII 2 (d2)(0i)(Si) (3)
§ 33. Comparison of the Primitive Sentences
AND Rules of Language II with those
OF OTHER Systems
1. The method of giving schemata of primitive sentences instead
of stating the primitive sentences themselves originated with von
Neumann [Beioeisth.^ and has also been applied by both Godel
[Unentscheidbare] and Tarski [Widerspruchsfr,].
2. Sentential calculus, Russell [Princ, Math,"] had five primitive
sentences ; these were reduced to four by Bernays [Aussagenkalkul],
Our system of three .primitive sentences PSII 1-3 is due to
Lukasiewicz [Aussagenkalkul],
3. Functional calculus. By this is usually understood a system
which corresponds approximately to our rules PSII (1-3), 16-19,
and RII I and 2. We will now compare these rules with the corre-
sponding ones in a number of other systems, with the object of
showing briefly that to the primitive sentences and rules of the
other systems which are not amongst those of Language II corre-
spond (on the basis of a suitably chosen translation) demonstrable
syntactical sentences about demonstrability and derivability in II.
Thus for all demonstrable sentences of the other systems, there are
corresponding demonstrable sentences in 11 ; and for every relation
of derivabiUty in one of the other systems, there is a corresponding
relation in 11. In the earlier systems (not only in those which are
mentioned here) substitution with arguments was for the most part
§ 33 - COMPARISON OF II WITH OTHER SYSTEMS 97
admitted and undertaken in practice; apparently, however, exact
rules for carrying it out (see p. 90) have never been stated.
{a) Russell {[Princ. Math.] * 10, second version of the calculus of
functions) gives PSII 16 as a primitive sentence (* lo-i ( jc)(F(x))D
F(y)*^ and not as a schema. This necessitates a rule of substitution
which, however, is not formulated but merely tacitly applied.
Further, PSII 19 is given as a primitive sentence (* 1012), PSII 18
as a definition (*io oi), and RII i and 2 as rules
For our Theorem 32.3 6 (♦ 1 1 *2) Russell requires a primitive sentence
(*11-07) which is not necessary in II.
(b) Hilbert [Logik]^ like Russell, states PSII 16 as a primitive
sentence and adds the necessary substitution rule (a). Hilbert’s
second primitive sentence corresponds to our Theorem 32.1a.
Hilbert gives three more rules: Rule (] 9 ) corresponds to RII i, the
rules (y) to Theorems 32.2a and b. PSII 18 is proved in Hilbert
(Formula 33a) and he obtains RII 2 as a derived rule (y*).
(c) Godel [Unentscheidbare] does not use the existential operator,
and therefore PSII 18 is not necessary. Gddel’s schemata of primi-
tive sentences III i and 2 correspond to PSII 16 and 19. RIJ
and 2 are laid down as rules of inference (definition of ^ diaeck’Con-
sequence *).
(d) Tarski [Wider spruchsfr.] does not erect primitive sentences
for the calculus of functions but only lays down rules of inference
(Def. 9 ‘consequence’). 9 (2) is a rule of substitution; substitution
with arguments is not admitted, so that PSII 17 disappears. 9(3)
corresponds to RII i; 9(4) and 9(5) to Theorems 32.2a and c
respectively. RII 2 is replaced by 9(5), and PSII 16 by 9(5) to-
gether with 9(2). Since he does not make use of an existential
operator, PSII 18 is unnecessary.
4. Arithmetic. Like Peano ([Formulaire] II, § 2) we take ‘ 0 ’ and
a successor symbol (‘ ‘ ’) as primitive symbols. We do not make use
of Peano’s undefined pr ‘number’, because Languages I and II are
co-ordinate languages and consequently all expressions of the lowest
type are numerical expressions. Therefore (i) and (2) of Peano’s
five axioms are eliminated. To his axioms (3), (4) and (5) correspond
PSII 13, 12, and 20. — On real numbers, see § 39.
5. Theory of aggregates. Since we represent aggregates or classes
by pr (compare § 37), sentences containing variables p correspond
to the axioms of the Theory of Aggregates. — (a) An Axiom of
Infinity (Russell [Princ. Math.] II, p. 203 ; Fraenkel [Mengenlehre]
p. 267, Ax. VII, p. 307) is not necessary in II; the correspbnding
sentence, C(x)( 3 y)(y = x')’)» demonstrable. The reason for this
is that, in Peano’s method of designating numbers, given a numerical
expression, an expression for the next higher number can be formed.
(On this point compare Bemays [Philosophie] p. 364.) — (b) To
Zermelo’s Asciom of Selection (Russell [Princ. Math.] I, pp. ff-
SL 8
98 PART III. THE INDEFINITE LANGUAGE II
and [Math, PM.]; Fraenkel Ax. VI, pp. 283 fT.) corresponds
PSII 21. — (c) PSII 22 is an Axiom of Extensionality (Fraenkel Def. 2,
p. 272; G6del [Unentscheidbare] Ax. V, i ; Tarski [Widerspruchsfr.]
Def. 7(3)). — (d) An Axiom of Reducibility (Russell [Princ, Math.'l I,
p. 55) is not necessary in II, because in the syntax of II only the so-
called simple classification of types, and not Russell's 'branched'
theory, is carried out (compare p. 86). — (e) An Axiom of Compre-
hension (akin to the Axiom of Reducibility) (von Neumann [Be-
weisth.] Ax. V, i ; G 5 del Ax. IV, i ; Tarski Def. 7 (2) ; it corre-
sponds approximately to Fraenkel's Axiom of Aussondernng, v,
p. 281) is not necessary in II, since, according to the syntactical rules
of definition, a pr" can be defined by every sentence having n free
variables, not excluding even the so-called impredicative definitions
(concerning the legitimateness of which, see § 44). — (/) Finally, let
us examine the axioms of Fraenkel which have not previously been
mentioned ([Mengenlehre] § 16). The Axiom of Determinateness
(Fraenkel Ax. I) is in II a special case of PSII ii. Fraenkel's Axioms
of Flailing, of Summation, of the Aggregate of Sub-aggregates, of
Aussondemng, of Replacement (II-V or V', and VIII) are not neces-
sary in Language II, because the aggregates (pr) postulated by these
axioms can always be defined. Predicate-functors for the general
formation of these aggregates can likewise be defined (compare the
examples ' sm ' and ' clsm ', pp. 86 f.).
C. RULES OF CONSEQUENCE FOR LANGUAGE II
§ 34fl. Incomplete and Complete Criteria
OF Validity
One of the chief tasks of the logical foundation of mathematics
is to set up a formal criterion of validity, that is, to state the neces-
sary and sufficient conditions which a sentence must fulfil in order
to be valid (correct, true) in the sense understood in classical
mathematics. Since Language II is constructed in such a way
that classical mathematics may be formulated in it, we can state
the problem as that of setting up a formal criterion of validity for
the sentences of Language II. In general, it is possible to dis-
tinguish three kinds of criteria of validity,
I. We may aim at discovering a definite criterion of validity —
that is to say, a criterion of a kind such that the question of its
fulfilment or non-fulfilment could in every individual instance be
decided in a finite number of steps by means of a strictly estab-
§ 34<*- CRITERIA OF VALIDITY 99
lished method. If a criterion of this kind were discovered we
should then possess a method of solution for mathematical problems ;
we could, so to speak, calculate the truth or falsehood of every
given sentence, for example, of the celebrated Theorem of Fermat.
Some time ago Weyl {[Philosophie] p. 20) asserted — ^without, how-
ever, giving a proof — “A touchstone of this kind has not yet been dis-
covered and never will be discovered.” And according to the more
recent findings of Gddel [Unentscheidbare] the search for a definite
criterion of validity for the whole mathematical system seems to
be a hopeless endeavour. Nevertheless, the task of solving this
so-called problem of resolution for certain classes of sentences
remains both an important and a productive one; and in this
direction many significant advances have already been made and
many more may be expected. But if we seek a criterion which
applies to more than a limited domain, then we must abandon the
idea of definiteness.
2. We may set up a criterion of validity which, although itself
indefinite, is yet based upon definite rules. Of this kind is the
method that is used in all modem systems which attempt to
create a logical foundation for mathematics (for example, the
systems of Frege, Peano, Whitehead and Russell, Hilbert, and
others). We shall designate it as the method of derivation or the
d-method. It consists of setting up primitive sentences and mles
of inference, such as have already been formulated for Language
II. The primitive sentences are either given as finite in number, or
they emerge by substitution from a finite number of schemata of
primitive sentences. In the rules of inference only a finite number
of premisses (usually only one or two) appear. The construction
of primitive sentences and rules of inference may be understood
as the definition of the term ‘directly derivable (from a class of
premisses) ’ ; in the case of a primitive sentence, the class of pre-
misses is null. It is usual to construct the rules in such a way that
the term ‘ directly derivable * is always a definite term ; that is to
say, that in every individual case it can be decided whether or not
we have an instance of a primitive sentence — or of the application
of a rule of inference, respectively. We have already seen how the
terms ‘derivable’, ‘demonstrable’, ‘refutable’, ‘resoluble’, and
‘ irresoluble ’ are defined on the basis of this d-method. Since no
upper limit to the length of a derivation-chain is determined, the
100
PART III. THE INDEFINITE LANGUAGE II
terms mentioned, although they are based upon the definite term
* directly derivable’, are themselves indefinite. It was at one time
thought possible to construct a complete criterion of validity for
classical mathematics with the help of a method of derivation of
this kind; that is to say, it was believed, either that all valid mathe-
matical theorems were already demonstrable in a certain existing
system, or that, should a hiatus be discovered, at any rate in the
future the system could be transformed into a complete one of
the kind required by the addition of further suitable primitive
sentences and rules of inference. Now, however, Godel has shown
that not only all former systems, but all systems of this kind in
general, are incomplete. In every sufficiently rich system for which
a method of derivation is prescribed, sentences can be constructed
which, although they consist of symbols of the system, are yet not
resoluble in accordance with the method of the system — that is to
say, are neither demonstrable nor refutable in it. And, in par-
ticular, for every system in which mathematics can be formulated,
sentences can be constructed which are valid in the sense of
classical mathematics but not demonstrable within the system.
In spite of this necessary incompleteness of the method of deriva-
tion (on this point, see § 6o</), the method retains its fundamental
significance ; for every strict proof of any sentence in any domain
must, in the last resort, make use of it. But, for our particular
task, that of constructing a complete criterion of validity for mathe-
matics, this procedure, which has hitherto been the only one
attempted, is useless ; we must endeavour to discover another way.
3. In order to attain completeness for our criterion we are thus
forced to renounce definiteness, not only for the criterion itself but
also for the individual steps of the deduction. (For a general dis-
cussion of the admissibility of indefinite syntactical concepts see
§ 45.) A method of deduction which depends upon indefinite in-
dividual steps, and in which the number of the premisses need not
be finite, we call a method of consequence or a c-method. In the
case of a method of this kind, we operate, not with sentences but
with sentential classes, which may also be infinite. We have
already laid down rules of consequence of this kind for Language I
(in § 14) and in what follows we shall state similar ones for Lan-
guage II. In this way a complete criterion of validity for mathe-
matics is obtained. We shall define the term "analytic’ in such a
§ 34^. CRITERIA OF VALIDITY 10 1
way that it ia applicable to all those sentences, and only to those
sentences, of Language II that are valid (true, correct) on the
basis of logic and classical mathematics. We shall define the term
‘contradictory* in such a way that it applies to those sentences
that are false in the logico-mathematical sense. We shall call Qi
L-determinate if it is either analytic or contradictory; otherwise we
shall call it synthetic. The synthetic sentences are the (true or false)
sentences about facts. An important point is that Language II in-
cludes descriptive symbols and hence also synthetic sentences.
As we shall see, this influences certain details in the form of the
definition of ‘ analytic *.
The following table shows which terms used in the two methods
correspond to one another :
d-terms c~terms
(depending upon the (depending upon the
method of derivation) method of consequence)
derivable consequence ,
demonstrable analytic
refutable contradictory
resoluble L-determinate
irresoluble synthetic
In every one of these pairs of terms with the exception of the
last, the d-term is narrower than the corresponding c-term.
The completeness of the criterion of validity which we intend
to set up, as opposed to that which is dependent upon a d-method,
will be proved by showing that every logical sentence of the system
is L-determinate, whereas, in accordance with what was said
earlier, no d-method can be so constructed that every logical sen-
tence is resoluble.
When Wittgenstein says [Tractatus, p. 164] : “ It is possible ... to
give at the outset a description of all ‘true* logical propositions.
Hence there can never be surprises in logic. Whether a proposition
belongs to logic can be determined**, he seems to overlook the in-
definite character of the term ‘ analytic * — apparently because he has
defined ‘analytic’ (‘tautology’) only for the elementary domain of
the sentential calculus, where this term is actually a definite term.
The same error seems to occur in Schlick {Fundament^ p. 96] when he
says that directly a sentence is understood, it is also known whether
or not the sentence is analytic. “ In the case of an analytic judg-
ment, to understand its meaning and to see its a priori validity are
one and the same process.” He tries to justify this opinion by quite
102 PART III. THE INDEFINITE LANGUAGE II
rightly pointii^f out that the analytic character of a sentence de-
pends solely upon the rules of application of the words concerned,
and that a sentence is only understood when the rules of application
are clear. But the crux of the matter is that it is possible to be clear
about the rules of application without at the same time being able to
envisage all their consequences and connections. The rules of
application of the symbols which occur in Fermat’s theorem can
easily be made clear to any beginner, and accordingly he under-
stand the theorem; but nevertheless no one knows to this day
whether it is analytic or contradictory.
§ 346. Reduction
Our procedure in laying down the consequence-rules for
Language I (§ 14) was first to define the term * consequence* by
means of the expansion of the rules of inference and then, with its
help, the terms ‘analytic’ and ‘contradictory’. In laying down
the consequence-rules for Language II, we shall, for technical
reasons, do just the reverse: first we shall define ‘analytic’
and ‘contradictory’ and then, with the help of these terms,
the term ‘consequence’. We shall frame these definitions in such
a way that, in spite of the different methods, within the domain of
those sentences of Language II which are at the same time sen-
tences of I, the c-terms concerning the two languages coincide:
if a sentence of 1 is analytic, or contradictory, or synthetic, or a
consequence of R^'in I, then the consonant sentence of II has the
same property in II.
In consequence of the richer structure of Language II, and
particularly of the occurrence of variables p and f and primitive
symbols ptb and fUb of infinitely many levels, the definitions of
the c-terms are considerably more complicated for II than they are
for I. By way of preparation for these definitions, we shall first lay
down rules for the reduction of sentences. By means of reduction,
every sentence of II is univocally transformed into a certain
(usually simpler) standard form. The rules of reduction RR 1-9 are
to be understood in this way: to any sentence under consideration,
the first of these rules whose application is possible must always
be applied. Thus, the order of the rules must be taken into account
(especially in the case of RR 9^). If the application of one of the
rules (even out of turn, except in the case of RR 9 e) leads from
§ 34*. REDDCTION IO 3
to S2» then ©i and ©2 — ^ easily be established — are always
mutually derivable.
Let ‘ ©1' designate any sentence in question. results from
means “ ©^ is transformed in such a way that the (proper or
improper) partial expression of ©1, ?Ii, is replaced by
RR I. Every defined symbol is eliminated with the help of its
definition. (In Language 11 , all definitions are explicit.)
RR 2. Construction of the conjunctive standard form :
a. (©2 3 Ss) • (Sa D ©2) results from ©2 = S3.
b. ^^©2^ ©3 results from ©23 ©3.
c. ©2 • ©8 results from (©2 V ©3).
d. '^©aV '^©a results from '^(©2«©3).
c. (S2VS3)*(S2V©4) results from ©2V (©3* ©4) or from
(S 3 -S 4 )VS 2 .
f. ©2 results from ^ ^ ©2.
RR 3. Disjunction and conjunction. Here disjunctions and con-
junctions not merely of two, but of many, terms are meant; for
instance, (©2 V S3) V ©4 or ©2V (S3V ©4) is called a three-termed
disjunction having the three terms ©2, ©3, ©4. The cancellation of
a term is understood to include the cancellation of the appertaining
symbol of disjunction — or of conjunction — and of the brackets
which thus become superfluous.
a. If two terms of a disjunction (or of a conjunction) are equal,
then the first is cancelled.
b. If ©2 is a disjunction (or a conjunction) of which two terms
have the form S3 and S3, then 91 (or ^ 91 respectively) results
from ©2. [ 9 t is ‘0 = 0’.]
c. If ©2 is a disjunction of which one member is 91 , then 91
results from ©2.
d. A term 91 of a disjunction is cancelled.
e. A term 91 of a conjunction is cancelled.
f. If ©2 is a conjunction of which one member is ^^ 91 , then
^ 91 results from Sg.
RR 4. Every limited 3 -operator is eliminated with the help of
PSII 9.
RR 5* Equations,
a. 91 results from 9 [i= 2 Ii.
3i = 82 results from 3 ^} = Sa*.
104
PART III. THE INDEFINITE LANGUAGE II
c. ^ 31 results from nu = 3i* or 3i' =
3b = 3i results from 3i = 3b-
RR 6. Elimination of the sentential variables f.
a. Let fi be the first free f in 3^; j results
from Si. ^
®' (^) '' ( ~ 9l)
RR 7. A K-operator is eliminated :
a. When it is limited, by means of PSII 14;
b. When it is unlimited, by means of PSII 15.
RR 8. Let 02 be a sentence with a limited universal operator
(3 i) 3 i(S 3).
a. Let 3i.not occur as a free variable in S^; Sg results from 02-
b. Let 3i be nu; results from Sj.
c. Let 3i have the form 3*1; (3i)32(®3)«®3
results
from Sg.
( 3 i) (32) (fi) (3 33) (3 34) (fi (nu, 34) = 34) ^ (fi ( 33 '> 34) =
fi(33.3i)')''~(fi(3i*32) = 30^63] results from 63. (This sen-
tence is equivalenj: in meaning to (jj) [(3i^3i)^®3]i see defini-
tion of * Grgl’ on p. 59, and the transformation described in §29.)
RR 9. Construction of the so-called standard form of the
functional calculus (see Hilbert [Logik\ p. 63). Only unlimited sen-
tential operators now occur as operators. Such an operator is
called an initial operator of Sj when either nothing, or only un-
limited operators occur before it in 0i (apart from brackets), and
its operand (apart from brackets) extends to the end of Sj.
a. 02 results from (t>i) (02) or from (3 x>i) (S2) if does not
occur as a free variable in S2.
b. Let the first operator variable in Si which is equal either
to another operator variable, or to a variable which occurs as a
free variable in Sj, be ©i. This operator variable together with all
variables which are bound by it (that is to say all variables ©i
which occur as free variables in its operand) are replaced by
§ 34 *. REDUCTION 105
variables which are equal to one another, but which are not equal
to the other variables occurring in
c* (3 ®i) (r' ^2) from ^{Vi) (Sg).
d- (Pi) ®2) results from ^(3 t)i) (Sa).
e. The first operator in Si that is not an initial operator,
together with the appertaining operand-brackets, is so transposed
that it becomes the last initial operator.
A sentence is called reduced when none of the rules of reduction
can be applied to it. The application of the rules to a sentence Si
always leads by means of a finite number of steps to the ultimate
form, namely to a reduced sentence ; this we call the reductum of
Si, and the syntactical designation of it is: ‘^S*.
Theorem 34b.i. Si and ^Si are always mutually derivable.
Theorem 34b.2. If Si is reduced, then :
A. Si has one of the following forms : i. (Di)(® 2) or (3 Di)(S2),
where Vi occurs as a free variable in S2 and where S2 has one of
the forms i to 9. — 2. '-^S2, where S2 has one of the forms 5
to 9. — 3. S2V S3, where each of the two terms has one of the
forms 2, 3, 5 to 9. — 4. S2«®3, where each of the two terms has
one of the forms 2 to 9. — 5. 3i = 32» where both 3 are 3i and
where at least one has the form d or e (see under B). — 6. 3 b = 3 *
—7. ^ri = ^r2.— -8. 5 ui = 3rU2.— 9. ^r( 9 Irg).— 10. 91 .— ii. ^ 91 ;
only in the case of this form does 91 occur as a proper partial
sentence.
B. Every 3 in Si has one of the following forms :
a. nu. — b. 3i*> where 3i has either the form a or the form b.
(a and b are St.) — c. 3 i'» where 3 i has one of the forms c, d or e.
— d 5. — e. 5 u ( 9 lrg). — Every 3 b ®i has the form c or e.
C. Every in Si is either an undefined ptb, or a p, or of the
form gu(9lrg).
D. Every gu in Si is either an undefined fUb, or an f, or of the
form gu(9lrg).
Theorem 34b*3« If Si is logical, reduced, and closed, then Si
has one of the following forms: i. 9li,9l2, ••• 5ln(®2)i where i,
= I to n) is either (n,) or ( 3 t),), and S2 contains no operators,
but does contain the free variables Ui, ...d„; 2. 91 ; 3. ^^ 91 .
Theorem 34b.4. If Si is logical and definite, then Si is
either 91 or /^9l.
I06 PART III. THE INDEFINITE LANGUAGE II
Theorem 34b.5» If by the application of a rule of reduction
(even out of turn, except in the case of RR 9 a sentence of the
form ©2- ®2 results from then is 91.
Theorem 34b.6. Every atomic (but not every molecular) sen-
tence is reduced (see p. 88).
Theorem 34b.7. If 6^ is reduced and contains no proper
partial sentence, no variable, and at the most one ^pr or one
^fu, but neither nor ^ for n > i, then is an atomic sentence.
§34 ^. Evaluation
We shall not define the term * analytic’ explicitly, but instead
we shall lay down rules to the effect that a sentence of a certain
form is to be called an analytic sentence when such and such
other sentences fulfil certain conditions — for instance, when they
are analytic. We must do this in such a way that this process of
successive, reference comes to an end in a finite number of steps.
We shall therefore proceed from a sentence to simpler sentences,
for instance from Si to or from a reduced sentence to
sentences which contain a lesser number of variables. If gi, for
example, occurs as a free variable in Si, then we shall call Si
analytic when and only when all sentences of the form
are analytic; thus we refer for instance from ‘Pi(jc)’ to the sen-
tences of the infinite sentential class Pi (0) ‘ Pi (0*) ‘ Pi (0»)
In this manner, the numerical variable is eliminated. In the
case of a predicate- or functor-variable, however, the analogous
method does not succeed; a fact which has been pointed out by
Godel. Let Si be, for example, *M(F)* (in words: “M is true
for all properties”). Now, if from Si we refer back to the sen-
tences ‘M(Pi)’, ‘M(P2)’, and so on, which result from Si by
substituting for ‘F’ each of the predicates of the type in question
which are definable in II, in turn, then it may happen that, though
all these sentences are true, ‘M(F)* is nevertheless false — in so
far as M does not hold for a certain property for which no predi-
cate can be defined in 11. As a result ot GodePs researches it is
certain, for instance, that for every arithmetical system there are
numerical properties which are not definable^ or, in other words,
indefinable real numbers (see Theorem 60^.1, p. 221). Ob-
§34^. EVALUATION 10 ^
viously it would not be consistent with the concept of validity
of classical mathematics if we were to call the sentence: ** All real
numbers have the property M an analytic sentence, when a real
number can be stated (not, certainly, in the linguistic system con-
cerned, but in a richer system) which does not possess this pro-
perty. Instead we will follow Godel’s suggestions and define
‘analytic’ in such a way that ‘M(F)’ is only called analytic if M
holds for every numerical property irrespective of the limited
domain of definitions which are possible in II.
Thus, in the case of a p, we cannot refer to substitutions but
must proceed in a different way. Let ‘F’ occur in Si as the only
free variable, a for instance. Then we shall not examine the
defined predicates of this type, but instead all the possible valua-
tions {Bewertungen) for ‘F*. By a possible valuation (syntactical
designation, B) for ‘F’ (i.e. a value assigned to ‘F’) we shall Kere
understand a class (that is to say, a syntactical property) of accented
expressions. Now if Bi is a particular valuation for.‘F’ of this
kind, and if at any place in Sj ‘F* occurs with Sti as its argument
(for example, in the partial sentence ‘F(0")’), then this partial
sentence is — so to speak — ^true on account of Bi, if Sti is an ele-
ment of Bi, and otherwise false. Now, by the evaluation of Si on
the basis of Bi, we understand a transformation of Si in which
the partial sentence mentioned is replaced by 91 if Sti is an ele-
ment of Bi, and otherwise by ^ 91. The definition of ‘ analytic ’
will be so framed that Si will be called analytic if and only if every
sentence is analytic which results from Si by means of evaluation
on the basis of any valuation for ‘F’. And Si will be called con-
tradictory when at least one of the resulting sentences is a con-
tradictory sentence. We shall lay down analogous rules for the
other p -types.
A valuation for a free ^fi^ will consist in a correlation by means
of which to every St an St is univocally correlated. In the case
of the evaluation of a sentence o^ » the basis of a certain valuation
Bi for fi, we shall replace a partial expression fi(Sti) by that St 2
which by means of Bi is correlated to Sti- We shall lay down ana-
logous rules for the other f-types.
Let pti be descriptive ; here a valuation of the same kind as for
a p is possible. Here also, Si, in which pti occurs, will be called
analytic if the evaluation on the basis of any valuation for pri
Io8 PART III. THE INDEFINITE LANGUAGE II
leads to an analytic sentence. In contradistinction to the case of
a p, however, Si will here only be called contradictory if the evalua-
tion on the basis of any valuation for pti leads to a contradictory
sentence. For, in the case of a p. Si means: “ So and so is true for
every property*’, and this is false if it does not hold for even one
instance. Here, in the case of the pti, however. Si means: “So
and so is true for the particular property expressed by pri“ where
we have a ptb and therefore an empirically and not a logically de-
terminable property; and this sentence is only contradictory —
that is to say, false on logical grounds — if there exists no property
for which Si is true.
On the basis of the foregoing considerations, we shall now pro-
ceed to lay down first the rules of valuation, VR, and then the
rules of evaluation^ E*vR. Later, in connection with these, we shall
formulate the definitions of ‘analytic* and ‘contradictory*.
Symbols to which a valuation can be assigned are called con-
valuable [bewerthare] symbols (syntactical designation, ‘b*). The
convaluable symbols in $i are all descriptive pti, and fUb, and
are also all 3, p, and f in those places where they occur as free
variables in 61.
VR 1. As the valuation for a convaluable symbol bi, any valua-
tion may be chosen which, in accordance with the following rules,
is of the same type as bj.
a. A valuation of the type 0 is an St.
b. A valuation of the* type t^ ,^2, . . . is an ordered «-ad of
valuations which belong to the types t^ to respectively.
c« A valuation of the type (/j) is a class of valuations of the
type ^1.
d. A valuation of the type {t^ : t^ is a many-one correlation
by means of which, to every valuation of the type t^^ exactly one
valuation of the type t^ is correlated.
VR 2. Let SjL be a reduced sentence without operators ; for all
b of Sj, let valuations be chosen according to VR i, and, in par-
ticular, let equal valuations be chosen for equal symbols. Then,
by the following rules, a univocally determined valuation results
for every partial expression in Si of the form 3» 3lrg, ^r, or gfu.
a. nu itself shall be taken as the valuation for nu.
b. Let Sti be the valuation for 3 i I then Sti' shall be taken as
§34^. EVALUATION IO 9
the valuation for 3i'- (Thus, as the valuation for an St, the St
itself is always to be taken.)
c. Let the valuations Si to Sn be assigned to the terms STj to
91,, of 9lrgi. Then the ordered n-ad Si,S 2 ,...Sn shall be taken
as the valuation of 9Irgi.
d. Let9Ii be an expression, 3i or gu, of the form gug (3lrgi) ;
and let the valuations Si and S 2 be assigned to 9lrgi and gu 2
respectively. Then that valuation which is correlated by Sg to the
valuation Si shall be taken as the valuation for 9Ii.
According to these rules, the valuation of an expression 9Ii is
always of the same type as 9li itself.
Examples, i . In connection with VR ia\ A S for a free 5 belongs
to the type 0, and is therefore an St, for example ‘O'*' *. — 2. In con-
nection with VR I c: A S for a ^pr', for example, for ‘F’ in ' F(x) \
belongs to the type (0), and is therefore a class of St, that is to say,
a syntactical property of expressions which only applies to accented
expressions. — 3. In connection with VR 1 6, c: A 93 for a ^pr’, for
instance, for *G* in ‘ G(jc,y,2r)*, belongs to the type (0,0,0) and is
therefore a class of ordered triads (or a three-termed relation) of
St — that is to say, a three-termed syntactical relation between
accented expressions. — 4. In connection with VR ic: A SB for a
2pr^, for example, for in * M(F)* belongs to the type ((0)), and
is therefore a class of classes of St. — 5. In connection with VR id:
A © for an ^fu-, for instance, for in ^f{x,y)=^z* belongs to the
type (0, 0 : 0), and is therefore a correlation by means of which an St
is univocally correlated to every ordered pair of St, and is therefore
a many-many-one syntactical relation between St. — 6. In con-
nection with VR 2 0, by c : Let the St ‘ O' * and ‘ 0 * be chosen as the
valuations for and respectively, in accordance with VR la.
Then, in accordance with VR 20, 6, c, the expression ‘O', 0, O'" ’ is
the valuation for 0'"’. — 7. In connection with VR2d: We
have already (p. 86) considered the sentence ‘sm(F, G)(x)*
(“jc belongs to the sum of the classes F and G”). Instead of ‘sm*,
we will now put a variable ‘m* of the same type ((0), (0) : (0)) :
* m(FyG){x)\ As an example of gu2(9lrgi), let us lake from this
the *m(FyG)\ which has the form ^P^) and is of the
type (0). Let the class of the St from ‘O' ’ to ‘0"" * be chosen as the
valuation of ‘F* (according to VR i c) and the class of the St from
‘0**** to ‘0""** as the valuation of ‘G\ Then, according to VR 2c,
the valuation ©i for 9lrgi (‘F, G*; type (0),(0)) is the ordered pair
consisting of the two aforesaid classes in the aforesaid order. For
gu, (i.e. *m*)f let ©2 be arbitrarily chosen. According to VR id,
©2 belongs to the type ((0),(0) :(0)) and is a correlation by means
of which a valuation of the type (0) is univocally correlated to every
valuation of the type ((0), (0) : (0)), and therefore also to © 1 . We will
110
PART III. THE INDEFINITE LANGUAGE II
assume that is chosen in such a way that the class of St from
‘O' * to ‘0'"" * is correlated to Si. (This would, for instance, corre-
spond to the constant ' sm * as a value for ‘ m ’.) Then according to
VR 2d, this class is the valuation for *m(F,G) \
Let Si be a reduced sentence without operators ; and let valua-
tions for all b in Si be chosen according to VR i and valuations
for further expressions be determined in accordance with VR 2 .
Then the evaluation of Si , on the basis of the valuations, consists
in the transformation according to the following rules of evalua-
don, EvR 1 , 2 . If a non-reduced sentence results from a trans-
formation, it must first be reduced and then transformed further.
EvRi. Let a partial sentence S 2 have the form ^r2(2lrgi);
and let the valuations for 9Irgi and ^12 be ®i and Sj* respectively.
If ®i is an element of ®2 then S 2 is replaced by 91; otherwise by
^^ 91 .
EvR 2. Let a partial sentence S 2 have the form 9 Ii= 2 l 2 , but
not 91; and let the valuations for 9Ii and 9 I 2 be ®i and ®2 re-
spectively. If ®i and ®2 are identical, S 2 is replaced by 91;
otherwise by 91.
Theorem 34 c«i. Let Si be a reduced sentence without opera-
tors. The evaluation of Si , on the basis of any valuations for the
b which occur, leads in every case, in a finite number of steps, to
the final result; this is either 91 or ^91. — For every Ub and n
occurring in Si we have a valuation. From these valuations there
results a valuation for every 3» ?lt9» and gu which occurs.
Thus every partial sentence of the form ^r(9lrg) is replaced
either by 91 or by ^ 91 ; and likewise every partial sentence of the
form 9li=9l2, since 9li and have the form 3 , or In
this way, we get a concatenation of sentences 91 by means of
symbols of negation, disjunction, and conjunction, from which,
by the application of RR 2 and 3 , either 91 or ^91 results.
§ Definition of ‘Analytic in II ' and
‘ Contradictory in II '
The definitions oi ^ analytic' and ^contradictory' with reference
to Language II are, as we have already mentioned, considerably
more complicated than they are with reference to Language I.
On the basis of the foregoing stipulations concerning reduction
Ill
§34^f. ‘analytic* AND ‘contradictory* IN II
and evaluation, these definitions for II can now be embodied in the
following rules DA 1-3. (‘A* and ‘C* are here used as abbrevia-
tions for “the necessary and sufficient condition under which
5li or Si is analytic** and “...contradictory**, respectively.)
DA 1. Definition of ‘ analytic * and ‘ contradictory ’ (in II) for
a sentential class 5 li. We distinguish the following cases.
A. Not all sentences of Sii are reduced. A (or C) : The class
of the reducta of the sentences of Hi is analytic (or contradictory,
respectively).
B. All sentences of 5 li are reduced and logical. A: Every
sentence of is analytic; C: At least one sentence of is
contradictory.
C. The sentences of Ri are reduced^ and at least one of them
is descriptive.
a. An open sentence occurs in 5 ti. Let 5^2 be the class which
results from fti by replacing every sentence S,- by ( ) (Si)
(see p. 94). A (or C): is analytic (or contradictory,
respectively).
b. The sentences of Ri are closed. A: For every sentence
S, of Ri the logical sentence is analytic that results from Si
by replacing every descriptive symbol by a variable of the
same type, whose design does not occur in Si, equal symbols
being replaced by equal variables and unequal symbols by
unequal variables. C: For the arbitrary choice of one valua-
tion for every descriptive S3rmbol occurring in Ri (the same
valuation being taken for equal symbols) there is at least one
sentence in Ri which is contradictory in respect of this valua-
tion (see DA 3).
DA 2. Definition of ‘analytic’ and ‘contradictory* (in II) for
a sentence Si-
A. Si is not reduced. A (or C): ^Si is analytic (or contradic-
tory, respectively).
B. Si is reduced and open. A (or C): () (Si) is analytic (or
contradictory, respectively).
C. Si is reduced, closed and logical.
a. Si has the form (Oi) (Sg). A: S2 is analytic in respect
of every valuation of 0i ; C : S2 is contradictory in respect of
at least one valuation of Ui.
1 12 PART III. THE INDEFINITE LANGUAGE II
b. Si has the form (3 Oj) (S2). A : S2 is analytic in respect
of at least one valuation of ©i ; C : S2 is contradictory in re-
spect of every valuation of Oj.
ۥ Si has the form 51 or A : Form 91 ; C : Form 91 .
D. Si is reduced, closed and descriptive. A (or C) : the class
{Si} is analytic (or contradictory, respectively).
DA 3. Definition of “analytic (or contradictory) in respect of
certain valuations” for a reduced sentence Si* (These terms only
serve as auxiliary terms for DA i, 2. ‘A— 93 i’ and ‘C— Si’ here
mean: “necessary and sufficient condition under which Si is
analytic (or contradictory, respectively) in respect of Si ”, where
is a series of valuations, namely, that consisting of one valuation
for each symbolic design 6 occurring in Si (hence not for the
bound variables).
A. Si has the form (1)2) (Ss)* A -Si: for every valuation
S2 for t)2> ®2 analytic in respect of Si and S2. C - Si : for at
least one valuation S2 for 1)2, S2 i® contradictory in respect of Si
and S2.
B. Si has the form (3 02) (©2)- A -Si: for at least one valua-
tion S2 for 021 ®2 ^® analytic in respect of Si and S2- C - Si : for
every valuation S2 for Og* ®2 ^® contradictory in respect of Si
and S2-
C. Si contains no operator. A -Si (or C-Si): The result
of the evaluation of .Si on the basis of Si is 91 (or ^ 91 , re-
spectively).
Let Si (or fti) be arbitrarily given ; and let it be asked whether
Si (or fti, respectively) is either analytic, or contradict ry, or
neither, i.e. synthetic. Then in the first place one and only one
of the rules DA is applicable (for DA 2 Ca--c this results from
Theorem 34^.3). If this rule is DA 2 Cc or DA 3 C the question
will be decided by means of this rule. Every one of the remaining
rules, on the other hand, will refer back univocally to a second
question concerning one or more other S or a ft. Thus for Si
or fti, the univocal result is a sequence of questions which is
always finite and which terminates with one of those two final
rules. For an arbitrarily given sentence or sentential class, a
sufficient and necessary criterion for ‘analytic’ — and likewise
for ‘contradictory’ — can be formulated on the basis of this
”3
§ 34^. ‘ ANALYTIC * AND ‘ CONTRADICTORY * IN II
sequence of questions. (An example of this is to be found in
the proof of Theorem 34 A. i.) These terms are thus univocally
defined for all cases by means of the rules DA. But there is
no general method of resolution for the individual questions, far
less for the whole criterion. The terms ‘analytic* and ‘contra-
dictory* are indefinite.
We have formulated the definition of ‘analytic* in a word-
language which does not possess a strictly determined syntax.
The following questions now present themselves, i. Can this
definition be translated into a strictly formalized syntax-language,
2. Can Language II itself be used as the syntax-language
for this purpose? Later we shall show (Theorem 60^.1) that
for no (non-contradictory) language S can the definition of
‘ analytic in S * be formulated in S itself as the syntax-langujtge.
Hence the second question must be answered in the negative.
On the other hand, the first question can be answered in the
affirmative provided that has adequate means at its disposal,
especially variables p and f of certain types which do not occur
in 11.
If we take as our object-language not the whole of Language II
but the single concentric regions (see p. 88), then for our syntax-
language we have no need to go outside the domain of II. It is
true that the concept ‘analytic in II„’ is not definable for any n
in II„ itself as syntax- language, but it is always definable in a
more extensive region lln+m (perhaps always in Iln+i). Hence
every definition of one of the concepts ‘ analytic in II„ * (for the
various n), and also every criterion for ‘analytic in II* with
respect to a particular sentence of II, is formulable in II as
syntax-language.
A certain point in the given definition of ‘ analytic in II * may
appear dubious. For the sake of simplicity we will consider the
corresponding definition of ‘analytic in IIj*. Let a language S
be used as a formalized syntax-language (for example, a more
extensive region of II, or II itself). Since in IIj free and un-
defined ^ptb occur, the definition of ‘analytic in IIj* (corre-
sponding to DA i Cb, 2 Ca) will contain phrases such as “for
every valuation for a ^p^...**; this, according to VRia and
VR iCy is the same as saying “for all syntactical properties of
accented expressions ...*’. Now what is meant by this phrase
1 14 PART III. THE INDEFINITE LANGUAGE II
and how is it to be formulated in the symbolic language S? If
we said instead merely “for all syntactical properties which are
definable in S then the definition of ‘analytic in IIi’ would
not effect what is required of it. For just as for every language
there are numerical properties which are not definable in it (see
p. io6), so there are also syntactical properties which arc not
definable in S. 'I’hus it might happen that the sentence ‘Sj is
analytic in II^ ’ was true (analytic) in the syntax-language S, and
yet false (contradictory) in a richer syntax-language S', namely
if the phrase, “for all definable syntactical properties...”, con-
tained in the criterion for that sentence, although valid for all
the properties definable in S, was not valid for a certain property
which is only definable in S'. 'Fhus the definition must not be
limited to the syntactical properties which are definable in S,
but must refer to all syntactical properties whatsoever. But do
we not by this means arrive at a Platonic absolutism of ideas,
that is, at the conception that the totality of all properties, which
is non-denumerable and therefore can never be exhausted by
definitions, is something which subsists in itself, independent
of all construction and definition? From our point of view,
this metaphysical conception — as it is maintained by Ramsey
for instance (see Carnap [Logizismus^ p. 102) — is definitely ex-
cluded. We have here absolutely nothing to do with the meta-
physical question as to whether properties exist in themselves or
whether they are created by definition. The question must
rather be put as follows: can the phrase “for all properties...”
(interpreted as “for all properties whatsoever” and not “for all
properties which are definable in S ”) be formulated in the sym-
bolic syntax-language S ? This question may be answered in the
affirmative. The formulation is effected by the help of a uni-
versal operator with a variable p, i.e. by means of ‘(F) for
example. (That this phrase has in the language S the meaning
intended is formally established by the fact that the definition
of ‘analytic in S' is formulated in the wider syntax-language
S2, again in accordance with previous considerations (pp. 106 f.),
not by substitutions of the pr of S, but with the help of valua-
tions.) This is correspondingly true for the valuations of higher
types in the wider language regions.
§ 34^. ANALYTIC AND CONTRADICTORY SENTENCES
”5
§ 34 ^. On Analytic and Contradictory
Sentences of Language II
Si (or fti) is called L-determinate if Si (or fti, respectively) is
either analytic or contradictory. Si (or is called synthetic if
Si (or 5 li, respectively) is not L-determinate, and therefore is
neither analytic nor contradictory.
Theorem 34e.i. {a) Si and ^Si are either both analytic, or
both contradictory, or both synthetic. — (/>) Likewise Si and
0 (Si). — (c) Likewise Si and {Si}.
Theorem 34e.2. (a) If Si is analytic, then ^Si is contra-
dictory. — (b) If Si is contradictory and closed, then -^Si is
analytic.
Theorem 34e.3. If every sentence of fti is analytic, then fti
also is analytic ; and conversely.
Theorem 340.4. A Rj is contradictory if and only if at least one
sentence belonging to it is contradictory. A SI5 can be contra-
dictory even if no sentence belonging to it is contradictory. (See
the remarks concerning Theorem 14.4.)
Theorem 34e.5« A closed sentence Si is anal)rtic (or contra-
dictory) if (but not only if) the truth-value table (§ 5) of Si, in respect
of partial sentences from which Si is constructed with the help of
symbols of negation and junction, always yields ‘T* (or ‘F\
respectively) for all admissible distributions of ‘ T * and ‘ F \ In
this connection, a distribution is called admissible if it always
assigns ‘ T * to an analytic partial sentence, ‘ F * to a contradictory
partial sentence and ‘ T ’ or ‘ F * to a synthetic partial sentence.
Theorem 34e.6. (a) Si V S^ is analytic if (but not only if) Si
or S2 is analytic. — (b) Si V S2 is contradictory if (and only if) Si
and ^2 contradictory.
Theorem 34e.7. SIj =?fi is always analytic.
Theorem 34e.8. Let Ri be a sub-class of ^2- (^) ^^2 is ana-
lytic, then fti is likewise analytic, — {b) If Ri is contradictory then
R2 is likewise contradictory.
Theorem 34e.9. If fti + R2 i® contradictory and 5 li is analytic,
then R2 is contradictory.
We have already seen that the concepts ‘demonstrable’ and
‘refutable’ do not fulfil the requirement that they constitute an
Il6 PART III. THE INDEFINITE LANGUAGE II
exhaustive distribution of all logical sentences (which also include
all mathematical sentences) into mutually exclusive classes. This
circumstance provided the reason for the introduction of the
concepts * analytic* and ‘contradictory*. We must now determine
whether such a classification is effected by these new concepts ; the
result of this test is given in Theorems lo and ii.
Theorem 34e.io. No sentence (and no sentential class) is at
the same time both analytic and contradictory. — A testing of the
single rules DA one by one shows that the conditions for ‘ analytic *
and those for ‘ contradictory * are mutually exclusive in every case
provided that they are mutually exclusive in that case to which
further reference is made. In the last stage, namely DA z Cc or
3 C, they are definitely mutually exclusive; and therefore they are
so in general. [In contradistinction to the analogous theorem
concerning ‘demonstrable* and ‘refutable*. Theorem lo does not
require the assumption that Language II is non-contradictory.]
Theoreta 34e.11. Every logical sentence is L-determinate, that
is to say it is either analytic or contradictory. (There is, however,
no general method of resolution.) — For the purpose of indirect
proof, let us assume that Sj were both logical and synthetic. Then
according to DA 2.4, would be both logical and synthetic;
and, according to DA 2 5 , ()(^Si) also would be both logical
and synthetic. Let this be 02* Then Sg would be logical, reduced,
and closed, and therefore, by Theorem 34^.3, it would have one
of the following forms: i. 9Ii?l2 ...2ln (S3 )> where
(i = I to n) is either (n,) or (3 n,), and S3 contains no operators ;
2. 91 ; 3. ^^ 91 . According to DAzCc, the forms 91 and ~ 9 l are
excluded here, since S2 is supposed to be synthetic. Hence Sg
would have the first-mentioned form. Then, in accordance with
DA 2 Ca and i, in respect of at least one series of valuations for
©1, ...t)n, ®3 must be neither analytic nor contradictory. The
evaluation of S3 on the basis of such a series of valuations would,
in accordance with DA 3 C, lead to a sentence which is neither 91
nor ^ 91 . But, by Theorem 34C.1, that is impossible.
According to Theorem ii synthetic sentences are only to be
found amongst the descriptive sentences.
Theorem 34e.i2. If a definite Si is analytic, then it is also
demonstrable. — (By DA 2 A, Theorem 346.4 and 346.1.) On the
other hand a definite Sb may be analytic without being demon-
II7
§ 34^. ANALYTIC AND CONTRADICTORY SENTENCES
strable. — Amongst the indefinite Sj there are analytic ones which
are non-demonstrable, also some of the simple form pti (ji), where
pti is a definite pri (compare the examples in § 36). In a case like
this, pti (fUi(nu)), where fui is any undefined fUb, is a definite
Sb which is analytic but not demonstrable.
Theorem 34e.i3. Every definite Sj is resoluble, that is to say,
it is either demonstrable or refutable. For this a general method of
resolution exists.
§ 34/. Consequence in Language II
T wo or more sentences are called incompatible with one another
if the class constituted by them is contradictory; otherwise they
are called mutually compatible.
A sentence is (in material interpretation) a logical consequence
of certain other sentences if, and only if, its antithesis* is incom-
patible with these sentences. Hence we define as follows : is
called a consequence of Sii in II, if fti-f (®i)} is contra-
dictory. @1 is called independent of 5li, if Sj is neither a conse-
quence of 5^1 nor incompatible with 5 li. We shall use the defined
terms not only in the case of a sentential class Sii but also in the
case of one or more sentences (as premisses). For instance, we
call S3 a consequence of Si and S2 if ®3 is a consequence of
It happens sometimes that Si is a consequence of an infinite
sentential class fti, without being a consequence of any proper
sub-class of 5^1. [Example, Let pti be an undefined ptb) fti be
the class of the sentences pti (Si), and Si be pti (31).] It is thus
essential that the definition of ‘ consequence *, as opposed to that
of ‘ derivable ’, should refer not only to finite but also to infinite
classes.
The concept ‘ consequence ’ is related to the concept * derivable *
as ‘analytic’ is to ‘demonstrable’; that is to say, it is more com-
prehensive, but on the other hand it has the disadvantage' of pos-
sessing a much more complicated definition and a higher degree of
indefiniteness. ‘ Derivable ’ is defined as a finite chain of the re-
lation ‘directly derivable’. ‘Consequence* might be analogously
defined as a chain of a simpler relation ‘direct consequence’.
Il8 PART III. THE INDEFINITE LANGUAGE II
' Analytic’ would then be defined as ‘ consequence of the sentential
null class’ and ‘contradictory’ as ‘sentence of which every sen-
tence is a consequence In this way the definitions for Language I
were previously formulated (§ 14). In the case of the definitions
just given for Language II we took a different course, and for the
sake of simplifying the technical process first defined ‘analytic’
and ‘ contradictory * and from them the term ‘ consequence *. The
question now is whether the term ‘ consequence ’ as so defined is
related to the terms ‘analytic* and ‘contradictory’ in the way
described ; that this is the case is expressed in Theorems 5 and 7.
Further, it must be shown that the relation ‘consequence’ pos-
sesses a certain kind of transitivity. This would be obvious in the
case of the first method of definition, but here the proof is not so
simple (Theorem 8).
Theorem 34f.i. If Si is an element of fti, then Si is a conse-
quence of ill. always a consequence of Si-
Theorein 34f.2. If fti is analytic, and Si a consequence of ili,
then Si is also analytic. — ili + {'^( ) (Si)} is contradictory;
therefore, by Theorems 34^.9 and 34^.1 r, '^()(Si) is contra-
dictory, and hence, by Theorem 34 e.2 Si analytic.
Theorem 34f.3« If Si is contradictory and a consequence of
ill, is also contradictory. — According to Theorem 34 e. i 6
and zb, ^() (Si) is analytic, and hence, by Theorem 34^.9, ili
is contradictory.
Theorem 34f.4. Let S2 be a consequence of Si ; if Si is ana-
lytic, then S2 is likewise analytic ; if S2 is contradictory, then Si
is likewise contradictory.
Theorem 34^5* If Si is a consequence of the sentential null
class, then Si is analytic; and conversely. — This follows from
Theorem 34^.2.
Theorem 34^6. If Si is analytic, then Si is a consequence of
every sentence; and conversely.
Theorem 34f.7. If 5 li (or Si) is contradictory, then every sen-
tence is a consequence of 5li (or of Si, respectively); and con-
versely. — By Theorem 34^.86. Converse by Theorem 3.
Theorem 34f.8. If S3 is a consequence of ilg, and every sen-
tence of il2 is a consequence of ilj, then S3 is a consequence of 5li.
Proof. Let 51 ^ be the class of the sentences () (^S,) for every
S,- of ill ; likewise ilg for every S,- of ilg; and let Sg be ( )
§34/- CONSEQUENCE IN LANGUAGE II II 9
Then S® and all sentences of ^4 and ftg are reduced and closed.
Let a series of valuations for the b (here descriptive symbols) of a
sentence or a sentential class be designated by ‘S * with the corre-
sponding suffix. Assumptions: I. 5^2 + 0(63)} is contradictory;
hence also 5^5 + {^®6}* 2. For every S, of ftj* + ) (S|)}
is contradictory; accordingly for every of + is
contradictory. Assertion: + (S3)} is contradictory; that
is to say, 5^4 + Se} is contradictory. This, according to DA i Cb,
means : for any choice of ®4 and Sg, either or a sentence of
is contradictory in respect of ®4 or ©g, respectively. For the
purpose of indirect proof, let us suppose the contrary, namely:
S4 and ©3 are given in such a way that neither ^S® nor any
sentence of is contradictory in respect of ©4+©g. Assumption
1 means : for any ©5 and ©g, either Sg or a sentence of Rg is
contradictory in respect of ©5 or of ©g, respectively. Assumption
2 means : for every (5^ of R^ in the case of any choice of ©4 and
©^, either 0, or a sentence of 5^4 is contradictory in respect
of ©, or ©4, respectively. Hence, on our supposition, on the one
hand, for any arbitrary ©5, a sentence of ftg, say S7, would be
contradictory in respect of ©5 ; and on the other hand, as for every
Sj- of ftg, so also for S7 in the case of an arbitrary ©7 (contained
in ©g), S7 would be contradictory in respect of ©7. But this is
impossible ; for since S7 is closed, S7 and ^ S7 cannot both be
contradictory in respect of the same valuation (see Theorem
34^.26).
Theorem 34f.9. (a) If < 5 x 0^2 is analytic, then ©2 is ^ conse-
quence of ©1- — (b) If ©I is closed and if ©.^ is a consequence of
©1, then ©1 D ©2 is analytic.
Proof of 9 a. For a closed ©^ the proof is simple. For an open
©1 the procedure is as follows. Since ©j D ©2 is analytic, ( ) ©^ V
©2) also is analytic; further, (Si)v() (©g) also (the proof
is too long to be given here). According to Theorem 34^.26 the
negation of the last-named sentence is contradictory; hence
0 (^2) is likewise contradictory; hence also the class
{()(Si), ~()(S2)}. and hence (Si, ~() (S*)}. Therefore S,
is a consequence of ©j.
Proof of 9 b. |©i, '^ ( ) (©2)} is contradictory, hence ©1 •
^0 (©2) is also contradictory. Since this sentence is closed,
according to Theorem 34^.26 its negation is analytic, and con-
120
PART III. THE INDEFINITE LANGUAGE II
sequently ^ Si V ( ) (Gg) is likewise analytic. Therefore, since Gi
is closed, ^SiV S2 is analytic, hence also Si 062.
Theorem 34 f.io. Si and (31) (Gi) are consequences of the
class of the sentences
Si
. — This corresponds to the rule
DC 2 for Language I (p. 38).
§ 34^. Logical Content
We call the class of the non-analytic sentences (of II) which are
consequences of Gi or 5 ti (in II) the content of Gi or 5 li, respec-
tively (in II). (For the reason for this definition see pp. 41 f.) Let
'equipollent^ and 'synonymous' be given definitions for II ana-
logous to those for I (see p. 42). These formally defined terms
correspond exactly to what is usually designated in material
interpretation as ‘ equivalent in sense \ or ‘ equivalent in meaning *,
respectively, so long as ‘equivalent in meaning* is understood
as “of equivalent logical meaning** and not as “designating
the same object**. In order that two object- (or number-)
designations ?li and %2 *^^7 synonymous, 3li=2l2 not only
must be true but must also be analytic. (See § 75, examples 6
and 7.)
We say that Gi or 5 li has null content if its content is the null
class. By total content, ,we understand the class of all non-analytic
sentences.
Theorem 34g.i. If two sentences are consequences of one
another, then they are also equipollent ; and conversely.
Theorem 34g.2. If two sentences are equipollent, then they are
synonymous; and conversely.
Theorem 34g.3. {a) If ^1=^2 analytic, then and ^2
synonymous. — (6) If and are synonymous, and if 2fi=2l2 is
a sentence, then this sentence is analytic.
Theorem 34g.4« If Gi = S2 is analytic, then Gi and G2 are
equ’pollent; and conversely.
Theorem 34g.5. If Si (or fti) is analytic, then Gi (or ftj,
respectively) has null content', and conversely.
Theorem 34g.6« If Gi (or Sii) is contradictory, then Gi (or S^i,
respectively) has total content', and conversely.
§34^. LOGICAL CONTENT 1 21
Theorem 34g*7* Si.S2«...S„ and {Si, S2, ... Sn} are equi-
pollent.
Theorem 34S*S« The content of a disjunction is the product
of the contents of the terms of the disjunction. — If the product
of the contents of several sentences is null (and consequently, ac-
cording to Theorem 8, the disjunction of the sentences is analytic),
we say that the sentences have mutually exclusive contents.
§ 34/1. The Principles of Induction and
Selection are Analytic
We shall now prove that the Principle of Complete Induction
and the Principle of Selection are both analytic. These principles
are included amongst the primitive sentences which were 'pre-
viously stated for Language II (PSII 20 and 21, §30). By the
example of the Principle of Induction, we shall show how the
criterion of whether a certain particular sentence is anal3rtic or not
is developed step by step by means of the DA rules. The proofs of
Theorems i and 2 are interesting because they involve a funda-
mental question: in each one of these proofs, there is used a
theorem of the syntax-language which corresponds with the
theorem of the object-language whose analytic character is to be
proved.
Theorem 34h.i. The Principle of Complete Induction (PSII 20)
is analytic.
Construction of the criterion. Let us call PSII 20 S^. The neces-
sary and sufficient criterion of the analytic character of Si may be
transformed in the following manner, each step being univocally
established by means of the DA rules. By DA 2 A the criterion is :
^^Si must be analytic. Let this be 62- We find S2 by means of
reduction :
[(~F(0)vF(*)vF(j-)).(~F(0)v ~F(*')vF(j))]’.
Further, according to DAaS: ‘(F )(3 *)(>’) [•••]’ must be ana-
lytic. Let this be S3. For this to be analytic by DA 2 Ca, S2
must be analytic in respect of every valuation for ‘ By DA 3 B :
for every valuation 95i for "F\ and for at least one valuation S2
*x\* (y) [...]* must be analytic in respect of Bi and ©3. By DA 3 A :
in the case of every 95i for for at least one valuation 952
122
PART III. THE INDEFINITE LANGUAGE II
*x\ and for every S3 for *y \ the operand which occurs in the
square brackets — let it be S4 — must be analytic in respect of
Si, S2, and S3. By DA 3 C: in the case of every Si for 'F\ for
at least one S2 for ‘ x \ and for every S3 for *y \ the evaluation of
®4 on the basis of Si, S2, and S3 must lead to In this way the
criterion is constructed.
Proof that the criterion is fulfilled. Let ©5 be ‘ V F(ap) V
F(y)\ and ©3 be * r^F{Q)y ^F(jc')VF(y)*; ©4 is then ©s^Se*
Si is of the same type as ^F\ i.e. of the type (0); and therefore,
according to VR i a and c it is a class of ©t. With regard to Si,
three cases are to be distinguished: i. The ©t ‘0* does not belong
to Si*, 2. ‘0* and every other ©t belongs to Si*, 3. '0^ belongs to
Si but an ©t exists — say ©ti — which does not belong to Si- —
I. In case i, the evaluation of ©4, independently of S2 and S3,
always leads to 51 . For here, in accordance with VR 2 a and EvR i ,
‘F(0)* is replaced by and thus ^ ^F(O)* leads to
from which, by reduction in accordance with RR 2/, 91 results.
Then, by RR 3 c, 91 results from ©5 and from ©3, and hence, by
RR 3 a, also from ©4. — 2. In case 2, ©4 independently of S2» for
any S3, leads to 91 . For, since every ©t belongs to Si, so also does
the valuation for ^y \ S3. Therefore, in accordance with EvR i,
the evaluation ol'F(yy leads to 91 . Thence, as before, ©5, ©3, ©4
all yield 91 . — 3. In case 3, it is possible to state, for any Si, a S2
such that the evaluation of ©4 leads independently of S3 to 91 .
Since, namely, ‘ 0 ’ belongs to Si, but ©ti does not, as step by step
we erase from ©ti a stroke ‘ we get an ©t.^ such that it belongs
to Si, while ©t2* does not. (In this inference, complete induction
is applied in the syntax-language.) Now let us take ©t2 as S2
(which, by VR i a, is an ©t). Then, in accordance with EvR i,
*F(xy will become 91 . By VR26, ©12* is the valuation for
Hence, according to E)vR i, ‘F(;ci)* becomes ^ 91 , and
hence becomes ^^^ 91 , from which we get 91 . And
hence, as before, ©5, ©g, and ©4 issue in 91 . — The criterion
is fulfilled in all three cases; and ©1 (PSII 20) is accordingly
analytic.
Theorem 34h.2. Every sentence of the form PSII 21 (Principle
of Selection) is analytic.
The proof is easy but too long to be given here in full. For the
sake of a fundamental question which is involved, we shall, how-
§34^. INDUCTION AND SELECTION ANALYTIC 1 23
ever, at least indicate its form. Let us assume that Sj a sentence
of the form PSII 21. is then:
(3 P2)(O7)(3P3)(3P4)(3t)8)(3P5)(P6)(3t)9)(t)l0)K^ [^ 2 ])
where 02 is :
(Pl(p4)VPl(P3)V ~Pl(p«)VpJt)9)).(...) (~P2(t)7)V
~ (Ps = P 4 ) V ~ Pi (p«) V ~ Pe (d,„) V ~ P5 (Oj j) V ~ p, (n^) V
~ Ps (®u) (®10 =
®2 is a conjunction with 30 terms, every term of which is a dis-
junction having 4 or 8 terms. Let S, (; = i to 1 1) be the valuation
for D,- or p,-, respectively. According to DA, Si is analytic if the
following condition is fulfilled : for every there is a S2 of a
kind such that, for every S,, there is a ©3, ®4, ©g, of a kind such
that, for every ©e, there is a ©g of a kind such that for every ©^q
and ©ij, the evaluation of Sg based on ©^ to leads to 91 . Let
©1 be given arbitrarily. We may classify the possibilities with
regard to ©4 as follows : ©^ is either null or it is not ; ©1 contains a
null class as an element or it does not ; there are two classes be-
longing to ©1 and having an element in common, or there are not.
Then it is easy to show that, in each one of these cases, the criterion
is fulfilled. Here we shall only examine the most important case,
namely the last : ©^ and the classes belonging to ©^ are not null
and no two of the classes belonging to ©^ have an clement in
common. Then — assuming that the Principle of Selection holds in
the syntax-language — there is a selective class of ©4, that is to say,
a class such that it has exactly one clement in common with every
class belonging to ©4. Let us take this selective class as ©3. Then,
as it is easy to show (classification of cases : ©g is either an element
of ©1 or it is not), the given criterion can be fulfilled in every case.
The Principle of Selection itself is used in the foregoing proof.
It must be noted, however, that this principle docs not appear here
as a sentence of the object- language, but as a sentence of the syntax-
language which we use in our syntactical investigations. It is clear
that the possibility of proving a certain syntactical sentence de-
pends upon the richness of the syntax-language which is used, and
especially upon what is regarded as valid in this language. In the
present case, the situation is as follows: we can work out in our
syntax-language S (for which we have here taken a not strictly
determined word-language) the proof that a certain sentence, Sj,
124 part III. THE INDEFINITE LANGUAGE II
of the object-language II is analytic, if, in S, we have a certain
sentence at our disposal, namely, that particular sentence of S
which (in ordinary translation) is translatable into the sentence
Si of II. From this it follows that our proof is not in any way a
circular one. An exact analogue holds for the application of the
Principle of Induction of the syntax-language in the proof of
Theorem i. The proofs of Theorems i and 2 must not be inter-
preted as though by means of them it were proved that the Prin-
ciple of Induction and the Principle of Selection were materially
true. They only show that our definition of ‘analytic* effects on
this point what it is intended to effect, namely, the characterization
of a sentence as analytic if, in material interpretation, it is regarded
as logically valid.
The question as to whether the Principle of Selection should be
admitted into the whole of the language of science (including also
all syntactical investigations) as logically valid or not is not decided
thereby. That is a matter of choice, as are all questions concerning
the language-form which is to be chosen (cf. the Principle of
Tolerance, § 17 and § 78). In view of our present knowledge of
the syntactical nature of the Principle of Selection, its admission
should be regarded as expedient. The fact that by means of its
admission the construction of the mathematical calculus is ob-
viously considerably simplified speaks for it. Against it, there is
hardly anything to be said, so long as the existence of any con-
tradiction in it has not been proved (and seems, on the contrary,
highly improbable).
§ 34/. Language II is Non-Contradictory
We have already attempted to represent the inexact concept of
logical validity (in II) by means of two different terms : the d-term
^demonstrable' and the c-term * analytic'. The relation subsisting
between these two terms must now be examined more closely.
We shall show that the second term is an extension of the first :
every demonstrable sentence is analytic, but not conversely. In
the same way we shall show that if Si is derivable from ftj. Si is
also always a consequence of ftj. In connection with this, we shall
show that Language II is nourcontradictory — ^that is to say, that
two sentences Si and ^ Si are never demonstrable in II.
§ 34 *. II IS NON-CONTRADICTORY 1 25
In order to show that every demonstrable sentence is analytic
(Theorem 21) we must prove that every one of the primitive sen-
tences PSII 1-23 of Language II (§ 30) is analytic. The individual
primitive sentences will be tested one after the other in the fol-
lowing paragraphs (Theorems 2-14).
Theorem 34i.i. All sentences which are demonstrable in the
ordinary sentential calculus — hence, for example, the Principk
of Excluded Middle^ the Principle of Contradiction, and the
Principle of Double Negation— are analytic. — This follows from
RR 2, 3.
Theorem 341.2. The primitive sentences PSII 1-6 are ana-
lytic. — ^This follows from Theorem i.
Theorem 341.3. The primitive sentences PSII 7-9 are ana-
lytic. — This follows from RR 83 , c, 4 and Theorem 343.5. ,
Theorem 341.4* Every sentence of the form PSII 10 is analytic.
— This follows from RR 5 a.
Theorem 341.5* Every sentence of the form PSII i i*is analytic,
— The proof is a simple one based upon a differentiation of cases :
and U2 either have or have not the same valuation.
Theorem 34I.6. PSII 12 is analytic. — This follows from
RR 5 c, 2/.
Theorem 341.7* PSII 13 is analytic. — This follows from
RR 2 3, 5 3, 3 3.
Theorem 34I.8. PSII 14 and 15 are analytic. — This follows
from RR 7 a, 3 , and Theorem 343.5.
Theorem 34I.9. Every sentence of the form PSII 16 is analytic.
Proof By (partial) reduction we get : (3 Ui) Sj V
This is analytic since the operand is analytic in respect of at least
one valuation, ®i, for inasmuch as 9li, or any arbitrary valua-
tion for 9 li, may be taken as ®i.
Theorem 341.10. Every logical sentence of the form PSII 17 is
analytic. — The primitive sentence PSII 17, the Principle of
Substitution with arguments^ represents one of the critical points
in the logico-mathematical system, especially in the case where
so-called surplus variables occur.
Proof, Let S3 be a logical sentence of the form PSII 17; and
let 3 lrgi be ...Ufc. We will assume that in addition to these
variables (which do not necessarily occur at all) S2 contains the
126
PART III. THE INDEFINITE LANGUAGE II
surplus free variables Vk+u •••Dm (by surplus variables are under-
stood those that do not occur in SlrQi). Let the variables which
occur as free variables in Sj, in addition to Pj, be ---Vp. In
order to show that is analytic, we will show that ^63 is anal3rtic
in respect of any given series ® of valuations for the variables
By partial reduction we get for ^63
^ [S4 V 65], wherein S4 is (g pj Si) and Sg is Si j •
Two cases may be distinguished:
1. Let there be a valuation Si for Pi such that is
analytic in respect of Si and S. Then, according to DA 3
(3 ‘s analytic in respect of S; hence, so also is ^S4,
and further ^Sa-
2. Let there be no valuation for Pi of the kind described. Then,
for every arbitrary valuation S, for Pi, ('^ Si) is not analytic in
respect of and S, and therefore, since it is logical, in accordance
with Theorem 34^.11, it is contradictory. Thus, ^Sj is analytic
in respect of St and S. Now, on the basis of the given valuations
S we will choose a certain valuation Si for Pi in the following
manner. According to VR i c, a possible valuation for Pi is a class
of possible valuations for ?Irgi : now let Si be determined by the
condition that a possible valuation S, for 9Irgi shall be an element
of Si if, and only if^ analytic in respect of S, and S.
Pi is always followed in Si by an argument-expression. Let a
certain partial sentence in Si containing Pi be Pi('ili,’2l2>
Assume that S' is the series of valuations for 9ti, which,
according to VR 2, result from the valuations S (of which here
only the valuations for the free variables occurring in 3Ii,
come into consideration); here, when partial sentences occur in
those expressions, we take 91 as the valuation for an analytic partial
sentence and as the valuation for a contradictory partial
sentence. Then ^^Pi ( 3 li, . . . 3 Ijfc), since it is logical, is either analytic
(Case a) or contradictory (Case b) in respect of Si and S'. In
Case a, according to EvR i, S' is an element of Si; in Case b it
is not. Now, S' is also a possible valuation for 2 lrgi. In Case a,
in accordance with our choice of Si, ^63 is analytic in respect of
S' (for Di, ...Ofc) and S; in Case b, it is contradictory. Thus, in
127
§ 34*. n IS NON-CONTRADICTORY
Case a,
is analytic in respect of ® , and in Case b
it is contradictory. — S5 is obtained from 0^ by replacing, at the
substitution-positions, a partial sentence of the form (3Ii, . . . 91^^)
by the corresponding partial sentence
As we
have already seen, any two corresponding partial sentences of this
kind arc either both analytic or both contradictory in respect of
and S. Hence, if ^0^ is analytic in respect of Si and S, then
^05 is also analytic in respect of S. It has been shown earlier
that ^01 is analytic in respect of ® and every arbitrary valuation
for Pi, and therefore it is also analytic in respect of S and Si.
Accordingly ^05 is analytic in respect of S, and hence so also
IS 03-
Theorem 34i.11* Every sentence of the form PSII 18 is
analytic. — This follows from RR9C, 2/, and Theorem 34^.5.
Theorem 341.12. Every sentence of the form PSII 19 is ana-
lytic. — By means of partial reduction, we get
the rest of the proof is analogous to that of Theorem 9.
Theorem 341.13. Every sentence of the form PSII 22 is
analytic.
Proof, Let 01 have the form PSII 22. '^0i is
(3O3) [ (Pl (» 3 ) V P2 K) V (Pl = Pz))* (~P 3 (D3) V ~ Pl(03) V (Pl = P2))].
f'or this to be analytic, there must exist for any arbitrary valua-
tions Si and S2 for Pi and Pg respectively, a valuation S3 for D3
such that the evaluation of the operand on the basis of these
valuations leads to 91. By means of a classification of cases, it is
easy to show that this condition is fulfilled.
Theorem 341.14. Every semence of the form PSII 23 is
analytic.
Proof. Reduction leads to :
(3 Ol) • • • (3 »n) [~ (f 1 (Ol. • • • On) = fa (Ol. • ■ • D») ) V (fi = fa) ]•
For this to be analytic, there must exist for any arbitrary valua-
tions for fi and ^2 ^ series of valuations for Uj, ... such that the
128
PART III. THE INDEFINITE LANGUAGE II
evaluation of the operand leads to 91 . It is easy to demonstrate
that this condition is fulfilled. If any arbitrary valuations for
and fa are given, then either they agree with one another or they
do not. In the first case, the second term of the disjunction, and
hence, the whole operand, becomes 91 . In the second case, we
take a series of valuations for Ui, . .. such that, with it, by means
of the valuations for and fa, two different valuations are corre-
lated. Then the first term of the disjunction becomes 91 and hence
the whole operand becomes 91 .
Theorem 34i.i5. Every logical primitive sentence of II is
analytic. — This follows from Theorems 2-14, 34 A. i and 2.
Theorem 34i.i6« If < 5 i is analytic, then is also analytic.
Theorem 34i.L7« Every primitive sentence of II is analytic. —
This follows from Theorems 15 and 16.
Theorem 34Li8. Every definition in II is analytic. — By RR i
and Theorem 34^.7.
Theorem 34i.i9« (a) ©a is a consequence of and D S2. —
(b) (n)(Si) is a consequence of Si.
Theorem 341.20. If, according to RII i and 2 (§ 3 1), S3 is directly
derivable from Si or from Si and S2, then S3 is a consequence
of Si or of Si and S2, respectively. This follows from Theorem 19.
Theorem 341.21. Every demonstrable sentence (in II) is analytic,
— From Theorems 17, 18 and 20, and Theorem 34/2. The con-
verse is not true (example: Theorems 36.2 and 5). (See the
second diagram on p. 185.)
Theorem 34I.22. If S^ is derivable (in II) from Si, Sa, ... Stn>
then Sn is a consequence of . — This follows from
Theorems 17, 18, 20, and 34/8.
Theorem 341.23. ^91 is not demonstrable in II. — This
follows from Theorem 21 and DA 2 Cc.
A language S is called contradictory if every sentence of S
is demonstrable in S ; otherwise it is called non-contradictory.
(See § 59.)
Theorem 34I.24. Language II (as the system of the d-rules
PSII 1-23 and RII 1-2) is a non-contradictory language. — This
follows from Theorem 23.
Hilbert set himself the task of proving “with finite means” the
non-contradictoriness of classical mathematics. What is meant by
§ 34 *. II IS non-contradictory 129
‘finite means* is not stated exactly in any work of Hilbert's which
has been published up to now (including [GrundL 1934]), but pre-
sumably what we call ‘definite syntactical concepts* is intended.
Whether with such a restriction, or anything like it, Hilbert's aim
can be achieved at all, must be regarded as at best very doubtful
in view of Coders researches on the subject (see § 36). Even in
the achievement of the partial results which are attainable, there
are very considerable difficulties to be overcome. The proof which
we have just given of the non-contradictoriness of Language II,
in which classical mathematics is included, by no means repre-
sents a solution of Hilbert's problem. Our proof is essentially de-
pendent upon the use of such syntactical terms as ‘analytic*,
which are indefinite to a high degree, and which, in addition, go
beyond the resources at the disposal of Language II. Hence, the
significance of the presented proof of non-contradictoriness must
not be over-estimated. Even if it contains no formal errors, it
gives us no absolute certainty that contradictions in the object-
language II cannot arise. For, sinc&the proof is carried out in a
syntax-language which has richer resources than Language II, we
are in no wise guaranteed against the appearance of contradictions
in this syntax^language, and thus in our proof.
§ 35. Syntactical Sentences which
Refer to Themselves
If the syntax of a language is formulated in that language itself,
then a syntactical sentence may sometimes speak about itself, or
more exactly, it may speak about its own design — for pure syntax,
of course, cannot speak of individual sentences as physical things,
but only of designs and forms. For instance, states : “ a sentence
of the design... is closed (or: open, demonstrable, synthetic, and
the like)"; and here itself possesses the design which is de-
scribed in it. For every syntactical property, it is possible so to
construct a sentence that it attributes to itself — whether rightly or
wrongly — just this property. We shall state the method of doing
this, since it leads to important consequences for the questions of
• the completeness of languages and the possibility of a proof of
non-contradictoriness. We have already formulated the syntax of
Language I in that language itself. In the same way the syntax
130 PART 111 . THE INDEPINTTB LANGUAGE II
of Language II can he formulated in II itself and to an even wider
extent, since in Language II indefinite syntactical concepts can
also be defined. Our further investigations will have reference to
Language II, but they can easily be transferred to Language I,
since in them we use only definite symbols of the kinds which have
already occurred in 1.
‘str(«)* means: “the which has the value n*\ [For
example, str(4) is the SNQt ’.] Regressive definition:
str(0) = reihe(4) (i)
str (»■) = zus [str (n), reihe (14)] (2)
Let any syntactical property of expressions be chosen — for
instance, * descriptive* or *non-demonstrable (in II)’. Let Si be
that sentence with the free variable (for which we will take the
term-number 3) which expresses this property [in the examples:
‘DcskrA(jif)*, ‘^BewSatzII(r,a?)*; compare p. 76]. Let S2 be
that sentence which results from Si if for * jc ’ * subst [jc, 3, str (jc)] ’
is substituted, [In the second example, S2 is *^BewSatzII
(r,subst[jc,3,str(jc)])’.] By means of the rule which has been
stated earlier (p. 68), the term-number for every defined symbol
is univocally determined. Thus, if S2 is given, the series-number
of S2 can be calculated ; let it be designated by ‘ b ’ (* b ’ is a defined
33). Let the SNgentence subst [b, 3, str (b)] be S3; thus S3 is the
sentence which results from S2 when the St with the value b is
substituted for "x\ It is easy to see that, syntactically interpreted,
Ss means that Ss itself has the chosen syntactical property.
We will explain this point by the example of the property * non-
demonstrable (in II)’. Here instead of ‘Sa’ we will write *©’.
[This sentence forms the analogue in II to the sentence con-
structed by Gddel \Unentscheidbare], the only difference being
that in it we use a free instead of a bound variable. ] Let b2 be the
series-number of the S2 (given above) of this example. str(b2)
is an ^St; to make the following discussion clearer, we will
indicate this 6t by (this St consists of *0* and bj accents
and is thus far too long for anyone to write out in full). Hence,
0ii..=b2. Let © be the sentence which has the series-number
subst [1)2, 3, str (ba)] (or subst [O'* ••, 3, str (O** ••)]). Hence, © is
the sentence which results from S2 if is substituted for
*«’; © is accordingly the sentence '/^BewSatzII(r, subst [0"««,
§ 35 * SENTENCES WHICH REFER TO THEMSELVES I3I
3, 8tr (0*1 ••)])*• way, we have determined the wording o
ffi. Syntactically interpreted, it means that that sentence whicl
has the series-number subst [0****,3,str(0'*--)] is not demon-
strable. But that sentence is ® itself. Thus (S means that (g it
not demonstrable.
Incidentally, it is to be noted that a sentence of descriptivi
syntax can refer to itself in an even more direct manner, namely
not merely to its design but also to itself as a physical thing con-
sisting of printer's ink. A sentence which occurs at a certain place
can, in material interpretation, mean that that sentence which
occurs at that place, i.e. itself, possesses such and such a syn-
tactical property. And here it is even easier than in the case oi
sentences of pure syntax to construct for every given syntactical
property a sentence which — whether rightly or wrongly — attri-
butes that property to itself. Suppose the property in question
is expressed by the pr ‘Q’; then the sentence ‘ Q [ausdr (b, a)] ’
means : “ The expression occurring at the positions a to a + b has
the property Q" (compare p. 78). \Example: At the places a to
a + 8 (indicated, say, by numbered positions on a piece of paper)
let the sentence Si *DeskrA[ausdr(8,a)]' occur. Syntactically
interpreted. Si means that the expression which occurs at the
places a to a + 8 is a descriptive expression. But this expression
is Si itself. Incidentally, Si is true (empirically valid) since Si
contains the fUb ‘ausdr’.]
§ 36. Irresoluble Sentences
We will now show (following Godel's line of thought [Unent-
scheidbare]) that the sentence (5 constructed in the preceding Section
is irresoluble in II.
We have built up Language II in such a way that the syntactical
rules of formation and transformation are in agreement with a
material interpretation of the symbols and expressions of II which
we had in view. [From the systematic standpoint, the converse
relationship holds: logically arbitrary syntactical rules are laid
down, and from these formal rules the interpretation can be
deduced. Compare §62.] In particular, the definition of
* analytic (in II)’ is so constructed that all those sentences and
only those sentences which are logically valid in their material
132
PART III. THE INDEFINITE LANGUAGE II
interpretation are called analytic. Further, in the construction of
the arithmetized syntax of I in I (D 1-125), we proceeded in such
a way that a sentence of this syntax — and hence a syntactically
interpretable, logical sentence, namely, an arithmetical sentence
of I — turns out to be true arithmetically when and only when on
a syntactical interpretation it is a true syntactical sentence. [For
instance: ‘ BewSatz (a, b) * is arithmetically true when and only
when a is the series-series-number of a proof in accordance with
the rules laid down, and b the series-number of the last sentence
in this proof.] Now let us suppose that in the same way the
arithmetized syntax of II is stated in II. [For instance, ' BewSatzII
(r^xY is defined so that it means: “r is an ssNproof of the ^Nsen-
tence x'\ Here, ‘BewSatzII* is a definite pr.] Then a syn-
tactically interpretable arithmetical sentence of Language II will
here be logically valid, and therefore also analytic, when and only
when, materially interpreted, it turns out to be a true syntactical
sentence. Thus we have here a shorter method (which is, because
of its clarity, easy to use) of proving with respect to certain Sj
(a proof which is otherwise very tedious) that they are analytic
(or contradictory) ; this proof arises from a non-formal considera-
tion of the truth or falsity of the sentence in question in its
syntactical interpretation. [In the above example : if we can show
that the ssNgentence-series a is a proof of the SNsentence b, it is
thereby demonstrated that the sentence ‘ BewSatzII (a, b) * is
analytic in II.]
© was the sentence ‘ ^BewSatzII (r,subst [...])’; for the sake
of brevity we will write here ‘subst [...]* instead of ‘ subst [0“ •
3, str(OH ••)]’- The serie&-number of © was subst [...].
Theorem 36.i« If Language II is non-contradictory, © is not
demonstrable in 11 . — Suppose that there were an ssNproof a of ©.
Then the sentence of II which means this, namely ‘ BewSatzII
(a, subst [...])’, would be true, and thus analytic, and, since it is
definite, also demonstrable. Now if © were demonstrable, so also
would be which is BewSatzII (a, subst [...])*. But
this sentence is the negation of the previous sentence. Thus II
would be contradictory.
Theorem 36.2. © is not demonstrable in II. — From Theorems i
and 341.24.
§36. IRRESOLUBLE SENTENCES I33
Theorem 36.3. ® is not refutable in II. — Suppose that ® were
refutable, and therefore (compare p. 94) ‘ ^(r)(^BewSatzII
(r,subst [...]))’ demonstrable. Then ‘(3r)(BewSat2lI(r,subst
[...]))’ would be demonstrable, and, by Theorem 34f.2i, ana-
lytic, and therefore true; that means that a proof for the sentence
with the series-number subst[...] would exist, and therefore for
ffi. But according to Theorem 2* this is not the case.
Theorem 36.4. ® is irresoluhle in II. — By Theorems 2 and 3.
Theorem 36.5. ® is analytic. — In syntactical interpretation,
® means the same as Theorem 2, is therefore true, and conse-
quently analytic. Thus ® is an example of an analytic but non-
demonstrable sentence of II (see diagram, p. 185). Every sentence
of the form ® where 3^ is \ is analytic and definite and
therefore, according to Theorem 34^.12, also demonstrable; but
the universal sentence ® itself is not demonstrable.
Let 2 Bjj be the closed sentence ‘(3 Af)(r)(^BewSat2:II(r,2:))’.
In syntactical interpretation it means that there exists in II a non-
demonstrable sentence and that therefore Language II is non-
contradictory.
Theorem 36.6. SSBu is analytic. — ^ 333 ] 1 is true, according to
Theorem 341.23.
Theorem 36.7. SBji is not demonstrable in 11 . — Theorem 7
can be proved by applying the proof given by Godel {[UnenU
scheidbare] p. 196). We will indicate the argument very briefly.
The proof of Theorem 36.1 can be effected by the means at the
disposal of Language II ; that is to say, the sentence SUjiD® is
demonstrable in II. Now were aUu demonstrable, then, ac-
cording to RII I, ® would also be demonstrable. But this, by
Theorem 2, is impossible. The non-contradictoriness of II cannot
be proved by the means at the disposal of II, SBu is a new
example of an analytic but at the same time non-demonstrable
sentence.
Theorem 7 does not mean that a proof of the non-contradic-
toriness of II would not be possible at all; indeed we have^already
indicated such a proof. The theorem means rather that this proof
is only possible with the resources of a syntax formulated in a
language richer than II. The proof which we stated earlier
makes a very essential use of the term ‘analytic (in II)*; but this
134 III- INDEFINITE LANGUAGE II
term (as we shall see later) cannot be defined in any syntax
formulated in Language II.
Corresponding results are true for Language I also: if is the
analogously constructed sentence to ® in I ^ BewSatz (r, subst
is analytic but irresoluble in Language I. Let 3 B| be a
sentence of I which approximately corresponds to the sentence
2Bu (such as ‘ BewSatz (r, c) \ where c is the series-number of
Then 2Bi is analytic but irresoluble in 1. The non-con-
tradictoriness of Language I (the non-demonstrability of some
sentences in I) cannot be proved by the means at the disposal of 1.
The fact that the non-contradictoriness of the language cannot
be proved in a syntax which limits itself to the resources of that
language is not due to any particular weaknesses in Languages I
and 11. This property, as Godel [Unentscheidbare] has shown, is
an attribute of a large class of languages, to which belong all the
systems known hitherto (and possibly all systems whatever) which
contain within themselves the arithmetic of the natural numbers.
(On this point compare also Herbrand [Nm-contrad,] pp. 5 f.)
D. FURTHER DEVELOPMENT OF LANGUAGE II
§ 37. Predicates as Class-Symbols
Frege and Russell both introduce class-expressions in such a
way that, from every expression which designates a property (for
instance, from a pr^ or from a so-called one-termed sentential
function — that is to say, a sentence having exactly one free vari-
able) a class-expression is constructed which designates the class
of those objects possessing the property in question. In Language
II we do not intend to introduce any special class-expressions; in
their place we use the predicates themselves. In what follows we
shall indicate how a shorter method of writing can be introduced
in which arguments and operators can, under certain circum-
stances, be left out. The result of this is a symbolism that is per-
fectly analogous to Russell’s symbolism of classes. A sentence in
this symbolism can be paraphrased in the word-language in terms
either of “properties” or of “classes”, as one wishes.
A property (or cla^s) is called null [leer] when it does not apply
§37. PREDICATES AS CLASS-SYMBOLS I 35
to (or contain) any object whatsoever ; and universal when it applies
to (or contains) every object. Thus our definitions are as follows:
Def. 37 . 1 . Leer^o)(F) = /^( 3 A:)(F(jp))
Def. 37^. Un(o) (^s (*)(/!'(*))
Analogous definitions can be framed for other types; the type
of the argument (here: ‘( 0 )’ for ‘F’) may be attached in the form
of a suffix, for instance:
Def. 37 . 3 - Lcer(o, o) ~ (3 *) Qy) (F(*,y))
Now with the help of the symbols of negation and of junction
we will form some combined ipt:
Def. 37^ (~F)(*) = ^F{x)
Def. 37.5. {F V G){x) = (F{x)yG{x))
Def. 37 * 6 . {F.G)(x)={F(x).G{x))
Corresponding definitions may be framed for any other types,
including many-termed pr. Analogous ^r can be constructed
with the help of the other junction-symbols ; they are, however,
seldom applied in practice.
We define the pr ‘ A* and ‘V* for the null property and for the
universal property as follows:
Def. 37.7. Ao(jc) = /-«^(jc = jc)
Def. 37.8. Vq (x) = (x = x)
Corresponding definitions can be framed for all the remaining
pr-types in which the designation of the type of the appertaining
is attached as a suffix.
Theorem 37.9. *(F=G) = (x) (F(x) = G (x))* is demonstrable
(with the help of PSII 22 and ii). — Analogously we now define
as follows:
Def. 37.10. (FcG) = (x)(F(x)dG(x))
Corresponding definitions may be framed for any two pr of the
same type, and therefore, specifically, also for many-termed pr.
[According to the previously stated syntax of Language II,
instead of ‘ F V G ’ we should write ‘ V (F, G) ’ or ‘ sm (F, G) where
‘sm' (as in the example, p. 86) is an fu of the type ((0),(0):(0))
or, in general, of the type ((0>(0*(0) ^ whatsoever.
And instead of ‘ Fc G *, we should write ‘ C (F, G) * or ‘ Sub (F, G) *,
where ‘Sub* is a pr of the type (( 0 ),( 0 )) (compare p. 86), or,
more generally, of the type ((t),(t)). But we will here write
‘ F V G * and ‘ Fc G ’ in order not to deviate too far from the usual
Russellian symbolism.] According to Theorem 9 and Def. 10,
136 PART III. THE INDEFINITE LANGUAGE II
for a sentence of the form (0i)(02)--- (tJn)(pri(Uif
pr2(0i, ... On)) we can always write pri = pr2; and for a sentence of
the form (Di)...(t)„)(pti(Di,...D„)Dptj(Di,...D„)) we can always
write pti C pr2. For this mode of symbolization without arguments,
two different translations into word-language are possible. For
instance, let ‘ P * and ‘ Q ’ be pr^ ; then we can translate ‘ Pc Q ’ as :
“The property P implies the property Q“, or, if we wish, as:
“the class P is a sub-class of the class Q“; correspondingly “sub-
relation when it is a question of many-termed pr. Further, we
can interpret the ‘ P V Q * when it is used without arguments as
the“ji/w of the classes P and Q§ **, and ‘P#Q* as the** product of
the classes P and Q“; analogously also the “sum** and “product
of relations** in the case of many-termed pr. ‘A* and ‘V* used
without arguments can be interpreted as **null class** and **uni~
ver sal class** (or as “null relation** and “universal relation**, re-
spectively). As an example of an application of the class symbolism,
the Axiom of Selection PSII 21 may be used (the p which occur
are to be taken from suitable types of at least the second order) :
[(Me -- Leer )• (F) (G) ([M(F) . M(G) . - Leer (F. G)] 0
(F= G))] D (3 H) (F) [M(F) 3 A 1 (F. H)]
Hereby ‘A1 * (“cardinal number i**) is to be defined as follows
(compare § 38 6) ;
Al(F) = (3*)(:y)(F(3^) = {y=*))
The mode of symbolization whose introduction is indicated in
the foregoing is completely analogous to RusselPs symbolism of
classes; the whole theory of classes and relations of the [Princ.
Math.] can easily be put -into this simplified form. But we shall
not go into this here, as it raises no further fundamental problems.
§ 38. The Elimination of Classes
The historical development of the use of class symbols in
modem logic contains several noteworthy phases, the examination
of which is fruitful for the study even of present-day problems. We
select for our consideration the two most important steps in this
development, which are due to Frege and Russell. Frege [Grund-
gesetze] was the first to give an exact form to the traditional dif-
ferentiation between the content and the extent of a concept. Ac-
cording to his view, the content of a concept is represented by the
sentential function (that is to say, by an open sentence in which the
§38. ELIMINATION OF CLASSES 1 37
free variables serve to express indeterminateness and not univer-
sality). The extent (for instance, in the case of a property concept,
i.e. of a one-termed sentential function, the corresponding class) is
represented either by a special expression containing the sentential
function, or else by a new symbol which is introduced as an ab-
breviation for this expression. An identity-sentence with class
expressions here means the coextensiveness of the corresponding
properties (if, for instance, ‘kj* and ‘kj* are the class symbols
belonging to the pr ‘Pi* and ‘Pj’, then ‘ki = k4* is equivalent in
meaning to ‘ (x) [Pi (ac) = Pj (x)] *). Later on, Russell proceeded in the
same manner. Following the traditional modes of thought, how-
ever, Frege made a mistake at a certain point; and this mistake was
discovered by Russell and subsequently corrected.
It was a decisive moment in the history of logic when, in the year
1902, a letter from Russell drew Frege*s attention to the fact that
there was a contradiction in his system. After years of laborious
effort, Frege had established the sciences of logic and arithmetic on
an entirely new basis. But he remained unknown and unacknow-
ledged. The leading mathematicians of his time, whose mathematical
foundations he attacked with unsparing criticism, ignored him. His
books were not even reviewed. Only by means of the greatest per-
sonal sacrifices did he manage to get the first volume of his chief
work [Grundgesetze] published, in the year 1893. The second volume
followed after a long interval in 1903. At last there came an echo —
not from the German mathematicians, much less the German philoso-
phers, but from abroad : Russell in England attributed the greatest
importance to Frege’s work. In the case of certain problems Russell
himself, many yeiirs after Frege, but still in ignorance of him, had
hit upon the same or like solutions; in the case of some others, he
was able to use Frege’s results in his own system. But now, when
the second volume of his work was almost printed, Frege learned
from Russell’s letter that his concept of class led to a contradiction.
Behind the dry statement of this fact which Frege gives in the
Appendix to his second volume, one senses a deep emotion. But, at
all events, he could comfort himself with the thought that the error
which had been brought to light was not a peculiarity of his system ;
he only shared the fate of all who had hitherto occupied themselves
with the problems of the extension of concepts, of classes, and of
aggregates — amongst them both Dedekind and Cantor.
The contradiction which was discovered by Russell is the anti-
nomy which has since become famous, namely that of the class of
those classes which are not members of themselves. In his Ap-
pendix, Frege examined various possibilities for a way out of the
difficulty, but without discovering a suitable one. Then Russell, in
an Appendix to his work [Principles] which appeared in the same
year (1903), suggested a solution in the form of the theory of types,
according to which only an individual can be an element of a class of
the first level, and only a class of the nth level can be an element of
PART III. THE INDEFINITE LANGUAGE II
138
a class of the n + ith level. According to this theory, a sentence of the
form or * is neither true nor false; it is merely meaning-
less. Later on Russell showed that this antinomy can also be so formu-
lated as to apply not only to classes but to properties as well (the
antinomy of ‘ impredicable see § 60a). Here, also, the contradiction
is eliminated by means of the rule of types ; applied to pr^ (as sym-
bols for properties) it runs thus : the argument of a ^pr can only be an
individual symbol, and the argument of an *^+'pr can only be an ”pr.
Now it is a very remarkable fact that Frege himself had already
made a similar classification of all sentential functions into levels
and kinds which also were arranged according to the kinds of their
arguments ([Grundgesetsie] Vol. i, pp. 37 ff.). In this he had done
important preliminary work for Russeirs classification of types.
But on two points — like traditional logic and Cantor’s Theory, of
Aggregates — he made errors, which were corrected by means of
Russell’s rule of ^pes. It is because of these errors that, in spite of
the perfectly correct classification of functions, the antinomies
arise. Frege’s first error consisted in the fact that in his system all
expressions (or more exactly, all expressions which begin with the
assertion symbol) are either true or false. He was thus obliged to
count as false, expressions in which an unsuitable argument was
attributed to some predicate. It was Russell who first introduced
the triple classification into true, false, and meaningless expressions
— a classification which was to prove so important for the further
development of logic and its application to empirical science and
philosophy. According to Russell, those expressions which have
unsuitable arguments are neither true nor false ; they are meaning-
less (in our terminology: they are not sentences at all). When this
first error of Frege is corrected, then the antinomy of the term
‘ impredicable ’ can no longer be set up in his system — for the de-
finition would have to contain the contra-syntactical expression
*F(F) \ The antinomy which relates to classes, however, can still be
constructed in his system. For Frege made a second mistake in not
applying the type-classification of the predicates (sentential func-
tions), which he had constructed with such insight and clarity, to the
classes corresponding to the predicates ; instead of that, he counted
the classes — and similarly the many-termed extensions — simply as
individuals (objects) quite independently of the level and kind of
the sentential function which defined the class in question. And
even after the discovery of the contradiction, he still thought that he
need not alter his procedure (Vol. ii, pp. 254 f.), because he believed
the names of objects and the names of functions to be differentiated
by the fact that the former have a meaning of their own while the
latter remain incomplete symbols which only become significant
after being completed by means of other symbols. Now, since Frege
held the numerals ‘ 0 ‘ 1 *, ‘ 2 *, etc., to be significant in themselves,
and since, on the other hand, he defined these symbols as class
symbols of the second level, he was compelled to regard class
§38. ELIMINATION OF CLASSES I 39
symbols, as opposed to predicates, as individual names. Today we
have the tendency to regard all the partial expressions of a sentence
which are not sentences in their turn as dependent ; and to attribute
independent meaning at most to sentences.
In order to define a cardinal number in Frege’s sense without
making use of classes, we have only to replace Frege’s class of pro-
perties by a property of properties (designated by a ®pr). It is re-
markable that Frege at an earlier stage expressed this view himself
([Grundlagen] 1884, p. 80, Note): “ I think that [in the definition of
* cardinal number*], instead of ‘extent of the concept’, we might
say simply ‘ concept *. But then two kinds of objections would be
raised : . . . . I am of the opinion that both these objections could be
removed; but that might lead too far at this stage.” Later he
apparently abandoned this view altogether. Then again — as it
appears when one looks back — Russell seemed to be very close to
the decisive point of abandoning classes altogether. While for Frege
it was important to introduce the class symbols as well as the pre-
dicates — since in his system they obey different rules — the whole
question had a different aspect for Russell. In order to avoid
Frege’s error, Russell did not adopt the class symbols as in-
dividual symbols but instead he divided them into types which
correspond exactly to the types of the predicates. But by this means
a quite unnecessary duplication was introduced. Russell himself
recognized that it was of no importance for logic whether “classes”
— ^that is to say, anything which is designated by the class symbols
— “really exist” or not (“no-class theory”). The further develop-
ment proceeded ever more definitely in the direction of the stand-
point that class symbols are superfluous. In connection with
Wittgenstein’s statements, Russell himself later discussed the view
that classes and properties are the same, but he did not as yet ac-
knowledge it (1925: [Princ. Math.]y 2nd edition of Vol. i). The
whole question is connected with the problem of the Thesis of
Extensionality (see § 67). Behmann [Logik] introduces the class
symbolism merely as an abbreviated method of writing, in which the
predicates are given without arguments; he insists, however, on
differentiating between extensional and intensional sentences, hold-
ing that this method of writing is only admissible for the former.
Von NeUxnann [Bezveistheorie] and Godel [Unentscheidbare] do not
even symbolically make any difference between predicates and the
corresponding class symbols ; in the place of the latter, they simply
use the former. The critique of Kaufmann ([Unendliche]^ [Berner-
kungen]) concerning Russell’s concept of class is also worthy of note.
But this criticism is really directed less against the Russellian system
itself than against the philosophical discussions by Russell and others
of the concept of class, which do not properly belong to the system.
We will summarize briefly the development which we have
just been considering. Frege introduced the class expressions in
140
PART III. THE INDEFINITE LANGUAGE II
order to have, besides the predicates, something which could be
treated like an object-name. Russell recognized the inadmissi-
bility of such a treatment, but, nevertheless, retained the class
expressions. The former reason for their introduction having been
removed, however, they are now superfluous and therefore have
been Anally discarded.
§ 380. On Existence Assumptions in Logic
If logic is to be independent of empirical knowledge, then it
must assume nothing concerning the existence of objects. For this
reason Wittgenstein rejected the Axiom of Infinity, which asserts
the existence of an infinite number of objects. And, for kindred
reasons, Russell himself did not include this axiom amongst the
primitive sentences of his logic. But in Russeirs system [Princ.
Math,] as, well as in that of Hilbert [Logik]^ sentences such as
*(3 x)(F(jc)V rwF(je))' and ‘(3 Jc) = and others like them,
in which the existence of at least one object is stated, are (logi-
cally) demonstrable. Later on, Russell himself criticized this point
{[Math, Phil,]^ Chap, xviii. Footnote). In the above-mentioned
systems, not only the sentences which are true in every domain,
independently of the number of objects in that domain, but also
sentences (for example, the one just given) which are true, not in
every domain, but in every non-empty domain, are demonstrable.
In practice, this distinction is immaterial, since we are usually
concerned with non-empty domains. But if, in order to separate
logic as sharply as possible from empirical science, we intend to
exclude from the logical system any assumptions concerning the
existence of objects, we must make certain alterations in the forms
of language used by Russell and Hilbert.
We may proceed somewhat as follows : No free variables are ad-
mitted in sentences and therefore universality can only be expressed
by means of universal operators. The schemata of primitive sen-
tences PSII 18 and 19 are retained (see § 30); PSII 16 and 17 are
replaced by rules of substitution: (Oi) (Si) can be transformed into
Si(^), and (Pi)(Si) into j • RH 2 disappears; but
certain other rules must be laid down instead. In the language thus
altered, when an object-name such as ‘a’ is given, ‘P(a)' can be
derived from ‘(Jif) (P(jc))*; and again, ‘( 3 J<f)(PW)* from ‘P(a)’.
§38fl. EXISTENCE ASSUMPTIONS IN LOGIC I 4 I
I’he important point is that the existential sentence can only be
derived from the universal one when a proper name is available ; that
is to say, only when the domain is really non-empty. In the altered
language, as opposed to the languages of Russell and Hilbert, the
sentence ‘ W (P(jc))d( 3 Jc) (P W)’ is not demonstrable without the
use of a proper name.
In our object-languages I and II, the matter is quite dif-
ferent owing to the fact that they are not name-languages but
coordinate-languages. The expressions of the type 0 here designate
not objects but positions. The Axiom of Infinity (see § 33, 5a) and
sentences like ‘( 3 ^) (x = x)* are demonstrable in Language II, as
are similar sentences in Language I. But the doubts previously
mentioned are not relevant here. For here, those sentences only
mean, respectively, that for every position there is an immediately
succeeding one, and that at least one position exists. But whether
or not there are objects to be found at these positions is not
stated. That such is or is not the case is expressed in a co-ordinate
language, on the one hand, by the fact that the fub at the positions
concerned have a value which appertains to the normal domain,
or, on the other, by the fact that they have merely a trivially
degenerate value. But this is stated not by analytic but by syn-
thetic sentences.
Example, In the system of the physical language, the sentence
which states that quadruples of real numbers (as quadruples of co-
ordinates) exist is analytic. In its material interpretation it means
that spatio-temporal positions exist. Whether something (matter or
an electro-magnetic field) is to be found at a panicular position is
expressed by the fact that at the position in question the value of the
density — or of the field-vector, respectively — is not zero. But
whether anything at all exists — ^that is to say, whether there is such
a non-trivially occupied position — can only be expressed by means
of a synthetic sentence.
If it is a question not of the existence of objects but of the
existence of properties or classes (expressed by means of predicates),
then it is quite another matter. Sentences like ‘( 3 ^)(^=^)*
(“There exists a property (or class)*') and ‘(3F) (Leer(F)y
(“ There exists a null property (or class) **) are true in every possible
domain, including the null-domain; they are also analytic and
logically demonstrable in the aforesaid system without existence
assumptions.
142
PART III. THE INDEFINITE LANGUAGE II
There are, however, also sentences about the existence of pro-
perties the legitimacy of which is disputed; the most important
examples being the Axiom of Reducibility and the Axiom of
Selection. We need not here go into the question of the Axiom of
Reducibility. In Russeirs form of language, it was a necessary
axiom on account of his branched classification of types (see
p. 86); but in Language II it is superfluous. [On the Axiom of
Comprehension^ which is closely related to it, see §33, 5^.] The so-
called Axiom of Selection (PSII 21) maintains the existence of a
selective class even in those cases where no such class can be
defined; and it is therefore a so-called pure (non-constructive)
existence statement. As such it is rejected by Intuitionism. In
Language II we have stated it as a primitive sentence, and we
regard the question of its assumption as purely one of expedience
(see pp- 97 f.)- That is true not only within the bounds of the
formalistic view of language as a calculus but also from the stand-
point of material interpretation. For, in such an interpretation,
only the atomic @5 given a meaning directly; the remaining
05 then acquire one indirectly. The Si (and with them all sen-
tences of mathematics) are, from the point of view of material
interpretation, expedients for the purpose of operating with the
S5. Thus, in laying down an Si as a primitive sentence, only use-
fulness for this purpose is to be taken into consideration.
§ 38A. Cardinal Numbers
In the material interpretation of Languages I and II, the 3 2re
to be interpreted for the most part as designations of positions or
of values of an fUb. Concerning the possibility of formulating
statements of cardinal numbers (** There are so and so many ...”)
we have so far said nothing. We will now proceed to show several
possibilities of doing so, which lie partly within and partly without
the syntactical framework set up for Language II.
The first method consists in defining every cardinal number
{Anzahl) as a ^r. For example, ‘A 5 (P)* (where ‘A 5 ’ counts as
one symbol) means: “The property P has the cardinal number 5,
that is to say there are exactly 5 numbers (positions) which have
this property.** Taking as an auxiliary term ‘Am 5 (P)* {MindesU
§386. CARDINAL NUMBERS I43
Anzahl; *Am 5 ’ is one symbol) which means: “There are at least
5 numbers which have the property P”, we deiine as follow^:
AmliF)=(2x)(F{x))
Am2(i?)s(3 (r^{x=y),F(x),F{y))
Am3(F)=(3x){2y)(3z) (,~{x=y), ^{x=z), ~{y=x),F{x)
.F(y).F(x))
and so on. On the basis of these minimum numbers, the exact
numbers are defined:
A 0 (F)= -^Aml(F)
A 1 (F)= (Aml(F)*^Am2(F))
A2{F)= (Asxa{F).^Am3{F))
and so on.
These definitions of the cardinal numbers correspond to those
of Frege and Russell; only here the second-level classes are re-
placed, for the reasons discussed in § 38, by second-level predi-
cates. These are here not written, as in Russell, simply as *0’,
*1’, and so on, because we already use these symbols in our
languages as symbok of the ty^ 0, and therefore may not use them
also as symbols of the type ((0)).
The second method employs special number-operators which
were not provided for in the previously stated syntax. Here, for
example, ‘(O'" 3 x) (P(Ar))’ means: “There are exactly 3 numbers
(or positions) having the property P.“ [*«* in ‘(tt3«)’ is not an
operator-variable and is not bound.] In this case we can either,
on the lines of the first method, de&e every individual number-
operator, or, more simply, construct two primitive sentences to
represent a general regressive definition:
(1) {03*)(F(*))=~(3*)(f’(*))
( 2 ) («i 3 *) (F{x))=(2x)(2 G) {G{y)~
~(:v=*)]).(« 3 *)(G(*))]
The third method expresses “There are 3 ...” by means of
‘Anz(3,P)’. As in the second method, analogous primitive sen-
tences can be constructed for the pr ' Anz’.
The fourth method is perhaps the most useful. It is like the first,
but in the place of a it uses a *fu, and writes ‘anz(P) = 3’.
As in the second method, two primitive sentences which take the
place of a regressive definition can be constructed for the functor
144 part III. THE INDEFINITE LANGUAGE II
‘anz* of the type (( 0 ) ; 0 ). But instead of the primitive sentences,
an explicit definition can also be constructed (according to the
method stated on pp. 88 f.):
anz(F)= (K»)(3/)(G) [([/(G)= 0 ] = ~(3 *)[G (*)]).
(«)(I/(G)=«'] =(3 *)(3 W*(>')(H( 3 ') = [G(>-).
-(:y=*)])-(/W=«)])-(®=/(^)]
In a precisely analogous way an ‘anz' of the type ((ii): 0 )
can be defined for the of the type {tj) and n> i.
A definite cardinal-number term referring to a limited domain
can similarly be introduced in accordance with the four methods
just given. The sentence: “There are 3 places up to the place 8
which have the property P*’ may be expressed, for example, as
foUows: I. ‘A3(8,P)*.~2. ‘(33^)8 (P(^c)) ’.- 3. ‘Anz(3,8,P)\
—4. ‘anz(8,P) = 3’.
All the cardinal-number terms which have been mentioned can
be applied to logical as well as to descriptive properties (for
example, to the number of the prime numbers less than 100, as
well as to the number of red positions).
§38^. Descriptions
By a description we understand an expression which (in material
interpretation) does not designate an object (in the widest sense)
by a name, but ctiaracterizes it univocally in a different way,
nafnely, by means of the statement of a property which belongs
only to that object.
Examples. Description- of a number: “The smallest prime num-
ber which is greater than ao ” ; of a thing : “ The son of A of a pro-
perty : “ The logical sum of the properties P and Q In the word-
language a description is effected by the use of the definite article in
the singular number (“the so-and-so”).
Profiting by the attempts of Frege and Peano, Russell has pro-
duced a detailed theory of descriptions: [Princ. Math.] Vol. i,
pp. 66 ff. and 173 ff. ; and [Math. Phil.].
, Following Russell’s method one could (in an extension of the
syntax of Language II) symbolize a description with the help of
a special descriptional operator ‘ix\ “That number (or position)
which has the pjoperty P” would then be written as follows:
*(7x)(P(x))’. We call a description of this kind an empty or a
§ 38 c. descriptions 145
univocal or an ambiguous description, respectively, if there is no
number, or exactly one number, or several numbers having the
property. A numerical description is used like a 3» for example
as an argument. ‘Q [0 Jf) (P(Af))]* means: ‘‘The number having
the property P has also the property Q.'* This sentence is to be
taken as true when, and only when, the description is univocal and
the described number has the property Q. It is obviously neces-
sary to make clearly recognizable the partial sentence (narrower or
wider) which is to express the property to be ascribed to the de-
scribed object. This can be done (as by Russell) by means of an
auxiliary operator: the whole description (consisting of descrip-
tional operator and bracketed operand) is put in square brackets
in front of the partial sentence in question. In accordance with
the material interpretation previously given, we can now construct
the following schema of primitive sentences which applies to
descriptions of any type whatsoever ( 3 , ^r, or JJu) :
[0 Di) (pti (Oi))] [pt* [(» Oi) (pti (Pi))] ] S [(13 Ol) (ptl (Di))»
(Di) (ptl(»l)3pt4(»l))]
The necessity for the use of the auxiliary operator may be seen
by a comparison between the following two sentences [analogy:
the necessity of the universal operator in order to be able to dif-
ferentiate between ( 31 ) and '^(Si) (Si)]:
[(,*)(PW)][~Q[(,*)(PW)]] (,)
-[0*)(PW)][Q[(,*)(P (*))]] (2)
(i) means: “There is exactly one P-number, and every P-number
(and therefore this one) is not a Q-number*'; (2), on the other
hand, means: “It is not true that there is exactly one P-number
and that every P-number is a Q-number.“ If the description is not
univocal (that is to say, if there are either no P-numbers at all or
several P-numbers) then (i) is false but (2) is true. To simplify
the symbolism it is possible (as Russell does) to rule that the clumsy
auxiliary operator may be left out when its operand is the smallest
partial sentence in which the description in question occurs. In
this case, for instance in (2), we speak of a “primary occurrence*'
of the description; otherwise, for instance in (i), of a “secondary
occurrence**. According to this rule, (2), but not (i), may be
written briefly thus: ‘ [Q [(^x) (P(^))]]^
Descriptions are expressions of a special kind which cannot in
SL
zx
146 PART III. THE INDEFINITE LANGUAGE II
all cases be treated in exactly the same manner as the other ex-
pressions ( 3 i or |]hi) of the type concerned. While, for in-
stance, according to PSII 16, (ji) (ptj (3i)) 0 ptj ( 3 i) >8 true for every
ordinary 3i> it is not always true when a numerical description is
used for 3i* For example, the sentence (3i)(pt2(3i))oprj[(»3i)
(pCi(3i))] may be falsified on account of the fact that the descrip-
tion is not univocal. The sentence which here holds in its place is :
(81) (prs( 3 i ))3 [(i 3 81) (pri (8i))3pr2 [(’ 81) (pti (8i))]] ; this sentence
is demonstrable with the help of the schema of primitive sen-
tences already given.
If we wish to use definite descriptions, we must write the
descriptional operator with a limit; "0^)5 (P(x))’ then means:
“That number up to 5 which has the property P.**
The K-operator is a descriptional operator of a very special
kind; and the clumsy auxiliary operator is not necessary for its
use. The K-descriptions, since they are always univocal, can be
treated like ordinary 3 * This univocality is, however, only achieved
by laying down the convention that when no number exists which
has the property in question, the value of the description is zero.
Herein lies the disadvantage of the K-operator; however, it might
prove expedient in many cases. The K-operator itself is only ap-
plicable to numbers; nevertheless, with its help very often pr and
fu of higher levels can also be defined. Let and 'g ' be ; and
let ‘Q’ be a of the type ((0:0), (0:0)) (so that ‘Q(/,^)* is a
sentence). Suppose' that we wish to define the functor *k’ so that
‘k(^)* is equivalent in meaning to “that functor / for which
Q(/,^) is true”. The definition can make use either of an
ordinary descriptional operator (with an operator variable f):
k(^)=0/)(Q(/,^))
or else of a K-operator (with an operator- variable 3):
k(f) W=(K>')(3/) [Q(/.^). {y=fm
If the first definition is set up, then the defined symbol ‘k’ cannot
be used everywhere like an ordinary fu of the type in question ;
this disadvantage does not occur in the case of the second
definition.
§ 39 - real NUMBERS 147
§ 39. Real Numbers
The real numbers, together with their properties, relations, and
functions, can be represented within the framework of the given
syntax of Language II. If a particular (absolute) real number
consists of the integral part a and the real number b (<1), this
number can be represented by means of a functor ‘k’ which is
defined so that k(0) = a, and, for «>0, k(n) = 0 or 1 respectively,
according to whether at the nth place in the development of the
dual fraction of b, * 0 ’ or ‘ 1 * occurs. In order that the develop-
ment of the dual fraction may be univocal, we exclude those dual
fractions in which, from some point onwards, only ‘0* occurs.
The real numbers with sign (positive or negative) can be repre-
ented in a like manner.
The method of representation of real numbers indicated here was
stated by Hilbert [Grundlagen^ 1923] (see also von Neumann
[Beweisth.'l). Hilbert has planned a construction of the theory of
real numbers on this basis, but up to now he has not produced it.
A real number is thus represented by means of a of the
type (0:0); we shall designate this type briefly by ‘r\ Then a
property (or aggregate) of real numbers (for example, “algebraic”
or “transcendental” numbers) is expressed by means of a of
the type (r); a relation between two real numbers (for example:
“is greater than” or “is a square root of”) by means of a of
the type (r, r) ; a function of a real number (such as : “ square root ”
or “ sine ”) by means of a of the type (r : r) ; a function of two
real numbers (for instance : “ product” or “ power”) by means of a
of the type (r, r : r) ; and so on. The arithmetical equality of two
real numbers fUi and fu2 is expressed by fUi = fU2; for this sentence
(according to PSII 23 and ii) is true when and only when the
values of the two functors agree for every argument, and therefore
when and only when the two dual fractions coincide at all places.
As opposed to the equality of two natural numbers (represented
by St), the equality of two real numbers, even when they are
stated in the simplest possible form, is, in general, indefinite
— since it refers back to an unlimited universality. A complex
number is an ordered pair of real numbers, and thus an expression
of the type r,r; a function of one or two complex numbers is a
*fu of the type (r, r : r, r) or (r, r, r, r : r, r) respectively.
148 PART III. THE INDEFINITE LANGUAGE II
In this way all the usual concepts of classical mathematics
{Analysis^ Theory of Functions) can be represented, and all the
sentences which have been constructed in this domain can be
formulated. The usual axioms of the arithmetic of real numbers
need not be set up here in the form of new primitive sentences.
These axioms — and hence the theorems derivable from them — are
demonstrable in Language IL
It will now be shown very briefly how the most important logical
kinds which are distinguishable with respect to sequences of
natural numbers^ and therefore also with respect to real number Sy
can be represented by means of syntactical concepts. First we must
distinguish between a sequence given by means of a mathematical
law and one given by a reference to experience. In the representa-
tion by means of ^fu^, this difference is expressed by the difference
between fuj and fUb- Thus the term “sequence of free selections’*
{freie Wahlfolge) of Brouwer and Weyl is represented by the
syntactical term ‘ fUb *• The regular sequences can be divided into
those that are calculable (see Examples i a and b) and those that
are incalculable (Example 2). Syntactically this difference is
characterizable as the difference between definite and indefinite
fUi; for the former, by means of a fixed method, the value can be
calculated for any position; for the latter, in general, this is not
possible. In the case of sequences determined by reference to ex-
perience, we can differentiate further into: i. Analytically regular
sequences ; in the caSe of these, the reference to experience is not
essential, since it is equivalent in meaning to a certain mathe-
matical law (Example 3), — 2. Empirically regular sequences;
although the determination of these cannot be transformed into a
law, yet they have the same empirical distribution of values as an
analytically regular sequence — whether by chance (Example 4 a)
or in conformity with a natural law (Example 4^). — 3. Irregular
or unordered sequences; for these there is no mathematical law
which, even in a merely empirical way, they could possibly obey.
For an fUb fUj, these three kinds are to be characterized syn-
tactically in the following manner: i. There is an fuj fu2such that
fU2 is synonymous with fUi, and therefore such that fUi = fU2 is an
analytic sentence. — 2. There is an fuj fU2 such that fUi = fU2 is a
synthetic but at the same time scientifically acknowledged sen-
tence (that is to say, in Language II it is a consequence of scientifi-
§39- real NUMBERS I 49
cally acknowledged premisses ; in a P-language it is P-valid (com-
pare p. 184)). — 3. Condition 2 is not fulfilled. [For all three
concepts a further classification may be made according to whether
the mathematical law in question is calculable or not, that is to say,
whether the fuj concerned is definite or not. ] It is to be noted that,
in the definition of the concept of the unordered sequences, the
kind of laws which are to be excluded must be stated; or, more
exactly, in syntactical terminology, the rules of formation for the
definitions of the fuj which are to be excluded must be stated, for
example by means of reference to a certain language. [E.g., let a
sequence fUj be called unordered in relation to Language II if
there is no fuj fU2 definable in II such that fUi = fU2 is valid in a
non-contradictory language which contains II (Example 5).] The
same holds good for the term “ irregular collective ** in von Mises’s
Theory of Probability.
Examples, i. Calculable regular sequences : (a) The recurring dual
fraction with the period ‘on*; (b) the dual fraction for tt. —
2. Incalculable regular sequence ‘ki*: let ki(n) be equal to i if a
Fermat equation with the exponent n exists; and otherwise let
ki(w) be equal to 0. — 3. Analytically regular sequence ‘k2*: let k3(n)
be equal to m if the nth cast of a certain dice shows an m ; our de-
finition is : kg (n) = ka (n) + 2 — (n), according to which the f iib ‘ kg *
is synonymous with the fui ‘k4* whose definition is: k^(fi)=^z . —
4. Empirically regular sequences : (a) Let k^ (n) be the number turned
up at the nth throw, where, however, whenever the dice falls, it
shows by chance alternately either a 3 or a 4. (Of course, this can
never be completely established, but it is conceivable as an assump-
tion.) (b) Let k3(n) be equal to i when a certain compass-needle,
used as a roulette pointer, in the position of rest after the nth play
points to the South, and equal to 2 when it points to the North.
According to natural laws, kfl = k4 is valid. — 5. Sequence ‘k7*,
unordered in relation to Language II: let k7(n) be equal to i when
n is a series-number of an analytic sentence of II, and otherwise
equal to 0. Since ‘analytic in II* is not definable in II (see p. 219)
there is no fui in II which has the same distribution of values as k7.
§ 40. The Language of Physics
Since, in Language II, not only logical but also descriptive
symbols (pr and fu) of the various types may occur, there is a pos-
sibility of representing physical concepts, A physical magnitude
(of a state or condition) is an fUb; the argument-expression con-
150 PART III. THE INDEFINITE LANGUAGE II
iains four real numerical expressions, namely, the time-space
co-ordinates; the value-expression contains one or more real
numerical expressions (for instance, in the case of a scalar, one;
in the case of an ordinary vector, three). A set of four co-ordinates
is an expression of the type r, r, r, r; we will designate this type in
a shorter way by ‘ q ’. [Examples: i . “ At the point kj, kg, kj, at the
time the temperature is k5 ” may be expressed e.g. as follows :
‘temp(ki,k2,k3,k4) = k5’, where ‘temp’ is a of the type (q:r).
2. “At the space-time point ks, k4 there is an electrical field
with the components ks. k^.ky” may be expressed, say, by
‘el(ki,k2,k3,k4) = (k5,k4,k,)’, where ‘el* is a of the type
(q:r,r,r).]
An empirical statement does not usually refer to one individual
space-time point, but to a finite space-time domain, A domain of
this kind is given by means of a of the type (q) — namely, by
means of a mathematical (prO or a physical (ptb) property which
belongs to all the space-time points of the domain in question and
only to those. A magnitude which is referred, not to individual
space-time points but to finite domains (for instance: tempera-
ture, density, density of charge, energy), can thus be represented
by means of a ®fub^ whose argument is a pr of the kind stated ; in
the case of a scalar, the type is ((q) : r) ; in the case where there are
several components, it is ((q):r, ... r). A property of a domain is
represented by means of a ^tb' of the type ((q)); the argument is
again the pr which determines the domain. The majority of the
concepts of everyday life, as well as those of science, are such
properties or relations of domains. [Examples: i. Kinds of things,
such as “horse”; “In such and such a place is a horse” means
“ Such and such a space-time domain has such and such a pro-
perty.” — 2. Kinds of substances, such as “iron”. — 3. Directly
perceptible qualities, such as “warm”, “soft”, “sweet”. — 4.
Terms expressing dispositions, such as “breakable”. — 5. Con-
ditions and processes of all kinds, such as “storm”, “typhus”.]
It follows from all these suggestions that all the sentences of
physics can he formulated in a language of the form of IL To this
end it is necessary that suitable fUb and ptb of the types given
should be introduced as primitive terms, and that, with their help,
the further terms should be defined. (Concerning that form of
the physical language in which synthetic physical sentences also —
§40. THE LANGUAGE OF PHYSICS I5I
for example, the most general laws of nature — are laid down as
primitive sentences, see § 82.)
According to the thesis of Phystcalisniy which will be stated later
(p. 320) but which will not be established in this book, all terms of
science, including those of psychology and the social sciences, can
be reduced to terms of the physical language. In the last analysis
they also express properties (or relations) of space-time domains.
[Exampks: “A is furious*’ or “A is thinking” means: “The
body A (i.e. such and such a space-time domain) is in such and
such a state ” ; “ The society of such and such a people is an economy
based on a monetary system” means: “In such and such a space-time
domain, such and such processes occur.”] For anyone who takes
the point of view of Physicalism, it follows that our Language II
forms a complete syntactical framework for science.
It would be a worth-while task to investigate the syntax of the
language of physics and of the whole of science in greater and more
exact detail, and to exhibit the most important of its .conceptual
forms, but we cannot here undertake such a thing.
PART IV
GENERAL SYNTAX
A. OBJECT-LANGUAGE AND SYNTAX-
LANGUAGE
We have now constructed the syntax of Languages I and II and
have thereby given two examples of special syntax. In Part iv we
shall undertake an investigation of general syntax — that is to say,
of that syntax which relates not to any particular individual lan-
guage but either to all languages in general or to all languages of a
certain kind. Before we go on, in Division B, to outline a getieral
syntax applicable to any language whatsoever, we shall first set
down, in Division A, some preliminary reflections concerning the
nature of syntactical designations and of certain terms which occur
in syntax.
§41. On Syntactical Designations
A designation of an object can be either a proper name or a
description of that object. The evident necessity of keeping in
mind the distinction between a designation and the object desig-
nated thereby (for instance, between the word ‘ Paris’ and the city
of Paris), although frequently emphasized in logic, is not always
observed in practice. If the object designated is such a thing as a
town, and the designation itself a word (either spoken or written),
the distinction is obvious. And for precisely that reason, in such
cases failure to differentiate between the two does not lead to any
harmful consequences.
If instead of “‘Paris’ is bi-syllabic” we write: “Paris is bi-
syllabic ’*, the method of writing is incorrect, because we are using
the word ‘ Paris ’ in two different senses ; in other sentences as the
designation of the city, and in the sentence in question as the
designation of the word ‘Paris’ itself. [In the second use, the
word ‘Paris’ is autonymous. See p. 156.] Nevertheless, in this
instance no confusion will arise, since it is quite clear that the sub-
ject here is the word and not the city.
>54
PART IV. GENERAL SYNTAX
It is another matter when the designated object is itself a
linguistic expression, as is the case with syntactical designations.
Here a failure to pay attention to the distinction leads very easily
to obscurities and errors. In meta-mathematical treatises — the
greater part of the word-text of mathematical writings is meta-
mathematics, and therefore syntax — the necessary distinction is
frequently neglected.
If a sentence (in writing) refers to a thing — my writing-table, for
instance — ^then in this sentence a designation of the thing must
occupy the position of the subject ; one cannot simply place the thing
itself — namely, the writing-table — ^upon the paper (this could only
be done in accordance with a special convention; see below). In
the case of a writing-table, and perhaps even of a match, this seems
self-evident to everyone, but it is not so self-evident when we are
dealing with things which are especially adapted to be put on paper,
namely, with written characters. For example, in order to say that
the Arabiofigure three is a figure, one often writes something of this
kind: *‘3 is a figure.** Now here, the thing itself which is under
discussion occupies the place of the subject on the paper. The
correct mode of writing would be : “ A three is . . . ** or ** * 3 * is, . . . **
If a sentence is concerned with an expression, then a desigriation of
this expression — namely, a syntactical designation in the syntax-
language — and not the expression itself, occupies the place of the sub-
ject in the sentence. The syntax-language may be either a word-
language or a symbol-language, or, again, a language composed of
a mbeture of words and symbols (for instance, in our text it con-
sists of a mixture of English words and Gothic symbols). The most
important kinds of syntactical designations of expressions are
enumerated below :
A. Designation of an expression as an individual, spatio-
temporally determined thing. (Occurs only in descriptive syntax.)
1. Name of an expression. [Occurs very seldom. Example:
“the Sermon on the Mount** (which can also be interpreted as a
description). ]
2. Description of an expression. \Example: “ Caesar *s remark
on crossing the Rubicon (was heard by so-and-so).**]
3. Designation of an expression by means of a like expression
§41- ON SYNTACTICAL DESIGNATIONS I S 5
in inverted commas. [Examples: “the saying ‘alca iacta est’”;
“the inscription *nutrimentum spiritus*.**]
B. Designation of an expressional design (see p. 15).
1. Name of an expsessional design (e.g. of a symbolic design).
[ExampUs: “A three”; “omega”; “the Lord’s Prayer”; “Per-
mat’s Theorem” (which can also be interpreted as a description);
”nu”;” 9 l”.]
2. Description of an expressional design by means of the state-
ment of a spatio-temporal position (indirect description, so-called
ostension, see p. 80). [Examples: “Caesar’s remark made at the
Rubicon (consists of three words)”; “ausdr (b, a)” (see p. 80).]
3. Description of an expressional design by means of syntactical
terms. [Examples: “The expression which consists of a three, a
plus symbol, and a four”; “(3i=3i)^*yj”: ]
4. Designation of an expressional design by means of an ex-
pression of this design in inverted commas. [Examples: ”‘3'”;
“‘3-1-4’”; “‘alea iacta est’ (consists of three words)”.]
C. Designation of a more general form (that is, a form that can
also apply to unequal expressions; see p. 16).
1. Name of a form (for instance, of a kind of symbol). [Ex-
amples: “variable”; “numerical expression”; “equation”; “n”;
“pr”;”3”]
2. Description of a form. [Examples: “ An expression consisting
of two numerical expressions with a plus symbol between them ” ;
“ 3 = 3 ”]
3. Description of a form by means of an expression of this form
in inverted commas together with a statement of the modifications
permitted. [Example: “An expression of the form 'x=y\ where
any two unequal variables may occur in the places of ‘jc’ and
It is frequently overlooked that the designation of a form' with the
help of an expression in inverted commas leads to obscurities if the
modifications permitted are either not given at all or are given in-
exactly. For instance, we often find : “For sentences of the form
‘(*) Fix))* so and so holds”, which leaves open such questions
as the following : Is it necessary for the [ ‘p ’ to occur in the sentence,
156 PART IV. GENERAL SYNTAX
or may any f occur in its place, or any sentence? Must the ip *F*
occur, or may any p take its place, or any pr ? Or, again, in the place
of ‘F(jc)* may we have any sentence with the one free variable *x\
or even with several free variables? This formulation is accordingly
obscure and ambiguous (quite apart from the fact that the inverted
commas are usually left out altogether, and 'that very often “for the
sentence...” is written instead of “for sentences of the form...”).
§ 42. On the Necessity of Distinguishing
BETWEEN AN EXPRESSION AND ITS DESIGNATION
The importance of distinguishing clearly between an expression
and its syntactical designation will readily be seen from such ex-
amples as the following; if, in the five sentences below, instead
of the expressions * cu *, “ \ ‘ omega *, ‘ ‘ omega * ‘ omega * * *,
we were in every case to use the word ‘omega', a very serious
confusion would ensue :
(1) is an ordinal type.
(2) ‘ cu * is a letter of the alphabet.
(3) Omega is a letter of the alphabet.
(4) ‘ Omega * is not a letter of the alphabet but a word of five
letters.
(5) The fourth sentence is not concerned with omega and there-
fore not with but with ‘omega’; hence in this sentence it is
not, as in the third sentence, ‘omega*, but “omega” which
occupies the place of the subject.
Since the name of a given object may be chosen arbitrarily, it is
quite possible to take as a name for the thing, the thing itself, or, as a
name for a kind of thing, the things of this kind. We can, for instance,
adopt the rule that, instead of the word ‘ match a match shall always
be placed on the paper. But it is more often a linguistic expression
than an extra-linguistic object that is used as its own designation.
We call an expression which is used in this way autonymous. In this
case the expression is used in some places as the designation of itself
and in others as the designation of something else. In order to
obviate this ambiguity of all expressions which also occur autony-
mously, a rule must be laid down to determine under what con-
ditions the first, and under what the second, interpretation is to be
taken. Example: We have used the symbols ‘ ‘ v’, * = ’, and so
forth sometimes as autonymous and sometimes as non-autonymous
symbols, but we have at the same time stipulated that they are
autonymous only when they occur in an expression containing
Gothic symbols (see p. 17). Counter-example: Formulations of the
§42. AN EXPRESSION AND ITS DESIGNATION 1 57
following kind are frequently found: ‘‘We substitute a + 3 for jc; if
a + 3 is a prime number,....** Here the expression ‘a -I- 3* is used
autonymously in the first case and non-autonymously in the second,
namely (to put it in the material mode of speech), as the designation of
a number. For this, no rule is given. The correct method of writing
would be: “We substimte ‘a + 3’ for ‘.v*; if a + 3 is a prime num-
ber,....” On the employment of autonymous designations in other
systems, see §§ 68 and 69.
Sometimes (even by good logicians) an abbreviation for an ex-
pression is mistaken for a designation of the expression. But the
difference is essential. If it is a question of an expression of the
object-language, then the abbreviation also belongs to the object-
language, but the designation to the syntax-language. The mean-
ing of an abbreviation is not the original expression itself, but the
meaning of the original expression.
Examples: If we write ‘Const’ as an abbreviation for ‘Con-
stantinople *, this abbreviation does not mean the long name, but the
city. If ‘ 2 ’ is introduced as an abbreviation for ‘ i + i *, then ‘ i + i *
is not the meaning of ‘2’, but both expressions have (in the
material mode of speech) the same meaning — that is (formally ex-
pressed) they are synonymous. An expression may be replaced in a
sentence by its abbreviation (and conversely), but not by its designa-
tion. The designation of an expression is not its representative, as
an abbreviation is. Very often obscurities ensue because a new
symbol is introduced in connection with a particular expression
without its being made clear whether this symbol is to serve as an
abbreviation or as a name for the expression. And sometimes the
confusion which results is impossible to eradicate, because the new
symbol is used in both senses, now in the word-text as a syntactical
designation, and now in the symbolic formulae of the object-
language.
Possibly many readers will think that, even though, strictly
speaking, it is necessary to distinguish between a designation
and a designated expression, yet the ordinary breaches of this rule
are harmless. It is true that this is often the case (for instance, in
the example given above of ‘ a + 3 ’), but the constant common dis-
regard of this distinction has already caused a great deal of con-
fusion. It is this disregard which is probably partly responsible
for the fact that so much uncertainty still exists concerning the
nature of all logical investigations as syntactical theories of the
forms of language. Perhaps the confusion between designation and
designated object is also to blame for the fact that the fundamental
158 PART IV. GENERAL SYNTAX
difference between the sentential junctions (e.g. implication) and
the syntactical relations between sentences (e.g. the consequence-
relation) is frequently overlooked (see § 69). Similarly, the ob-
scurity in the interpretation of many formal systems and logical
investigations may be traced back to this. *We shall come across
various examples of such obscurity later.
Frege laid special emphasis on the need for differentiating be-
tween an object-symbol and its designation (even in the witty but
fundamentally serious satire [Zahlen\), In his detailed expositions
of his own symbolism and of arithmetic, he always maintained this
distinction very strictly. In so doing, Frege presented us with the
first example of an exact syntactical form of speech. He does not use
any special symbolism as his syntax-language, but simply the word-
language. Of the methods mentioned above he uses for the most
par.t A3, B4, and C2 — expressions of the symbglism in inverted
commas, together with descriptions of forms with the help of the
word-language. He says {\GTundgesetze\f Vol. i, p. 4): “Probably
the constant use of inverted commas will seem strange ; but by means
of these I differentiate between the cases in which 1 am speaking
about the symbol itself and those in which I am speaking about its
meaning. However pedantic this may appear, I hold it to be neces-
sary. It is remarkable how an inexact method of speech or of writing,
which may have been adopted originally only for the sake of
brevity and convenience, with full awareness of its inexactitude, can
in the end confuse thought to an inordinate degree, once the con-
sciousness of its inaccuracy has vanished.”
The requirement laid down by Frege forty years ago was for a long
time forgotten. It is true that, on the whole, as a result of the works
of Frege, Peano, Schrfider, and particularly of Whitehead and
Russel] [Princ, Math.]^ an exact method of working with logical
formulae has been developed. But the contextual matter of nearly
all logical writings since Frege lacks the accuracy of which he gave
the model. Two examples may serve to indicate the ambiguities
which have arisen in consequence of this.
Example i. In the text of the majority of text-books and treatises
on logistics (Russell’s [Princ. Mat/i.j, Hilbert’s \Logik\y and Carnap’s
\LogistiK\ amongst them) a sentential variable is used in three or
four different senses: (1) As a sentential variable of the object-
language (as an [, for instance : ’). (2) As an abbreviation (and thus
a constant) for a compound sentence of the object-language (as a
constant [a, for instance: *A’). (3) As an autonymous syntactical
designation of a sentential variable (* f ’). (4) As a syntactical desig-
nation of any sentence (* 0 ’). Thus in many cases it is not possible
to arrive at the correct way of writing by merely adding inverted
commas. The usual formulation: “If p is false, then for any 9,
pOq is true ” cannot be replaced by “ If ‘ p ’ is false, . . . ” ; for ‘p ’ is cer-
§ 42 . an expression and its designation 159
tainly false (by substitution every sentence is derivable). We must
write either: “If ‘A* is false, then for any ‘B\ ‘AdB* is true^\
where ' A ’ and * B ’ are abbreviating constants of the object-language
(in this case with meanings left undetermined); or: “ If Si is false,
then for any S, the implication-sentence of Si and St is true.” If
suitable conventions are established (as on p. 17) then, instead of
“the implication-sentence of Si and Sj”, we may here write more
briefly: “ SiD S|.**
Example 2. In a treatise by a distinguished logician, the following
sentence occurs:
a is the formula which results from the
formula a when the variable x (if it occurs in a) is replaced throughout
by the combination of symbols p*' Here we are from the beginning
completely uncertain as to the interpretation. Which of the symbolic
expressions in this statement are used as autonymous designations,
and are accordingly to be enclosed in inverted commas if the cor-
rect mode of expressing the author’s meaning is to be achieved?
At first we shall probably be inclined to put ‘a’, * x \ and *p* in'in-
verted commas, and, on the other hand, to interpret ‘
a’ as a
syntactical mode of writing, and therefore not to enclose it as a
whole in inverted commas, but only its component letters :
(This would correspond approximately to our own
formula : ‘ 0 i ) ’ or, more closely, to : “ J , j.”) But the occur-
rence of the phrases “ the combination of symbols p ” and “ if jc occurs
in a ” rules out this interpretation ; for *p* is certainly no combination,
and obviously * x * does not occur in ‘ a *. Perhaps ‘ x * only is autony-
mous, while *p \ *a \ and ‘ which we should then have to
write taken as autonymous syntactical
designations? But opposed to this possibility is the circumstance
that in the symbolic formulae of the object-language which is dealt
with in the treatise, */) * and ‘ a * and even ‘ a * occur (for instance,
in the axiom *{x)aO a *). Possibly all the symbolic symbols and
expressions — not only in the sentences of the text but also in the
symbolic formulae of the system, are intended as non-autonymous
syntactical designations? In that case the way of writing that
sentence of the text was legitimate; and the axiom referred to
would correspond to our syntactical schema PSII 16. But, on the
other hand, this is not easy to reconcile with the rest of the text of
the treatise as it stands. We do not know to which object-language all
the formulae, as syntactical formulae, are to refer. For our context
here it is a matter of no importance which of these different in-
terpretations is intended. Our object is only to show what con-
i6o
PART IV. GENERAL SYNTAX
fusions arise when it is not made clear whether an expression belongs
to the object-language or is a syntactical designation, and, if the
latter, whether it is autonymous or not.
Frege*s demand for the maintenance of the distinction between a
designation and a designated expression is, as far as 1 know, strictly
fulfilled only in the writings of the Warsaw, school (Lukasiewicz,
Le^niewski, Tarski, and their pupils) who have consciously taken
him as their model. These logicians make use of special syn-
tactical symbols. This method has great advantages, although (as
Frege’s own example shows) it is not essential for correctness. The
clear symbolic separation of object-symbols and syntax-symbols does
not merely facilitate correct formulation, but, in the caseof the Warsaw
logicians, has been further justified by the fruitfulness of their in-
vestigations, which have led to a plenitude of important results. The
use of special syntactical symbols within the word-text ought, in the
majority of cases, to prove by far the most productive method ; for it
is both elastic and easily comprehensible, as well as sufficiently exact.
[This method is applied in the text of the present work : word-language
combined with Gothic symbols. The employment of Gothic letters
by Hilbert and of heavy print by Church are preliminary steps in
this direction.] In special cases, it may appear desirable to sym-
bolize completely the sentences and definitions of syntax and thus
to eliminate the word-language altogether. By this means an in-
creased exactness is attained, albeit at the cost both of facility in
treatment and of comprehensibility. Completely symbolized syn-
tactical definitions of this kind are used by Le^niewski and Godel.
In his [Neues System\ Le^niewski takes as object-language the sen-
tential calculus (with junction-variables in operators as well), and in
[pntologie^ the system of the €-sentences. As syntax-language, he uses
the symbolism of Russell, which, however, is only intended to serve
as an abbreviation for the word-language. Godel [Unentscheidbare]
takes as object-language the arithmetic of the natural numbers in a
modified form of the Russellian symbolism ; as syntax-language, he
uses the symbolism of Hilbert. (We have also applied this more
exact method in the formal construction of Part ii, where Language
I is at the same time both object-language and syntax-language.)
§43, On the Admissibility of Indefinite Terms
We have called a defined symbol of Language II definite when
no unrestricted operator occurs in the chain of its definitions;
otherwise, indefinite (§ 15). If pti is a definite ^pti then the propeny
which is expressed by means of pti is resoluble ; every sentence of
the form pti ( 9 lrgi) in which the arguments are definite 3 — in the
simplest case, accented expressions — can be decided according to a
fixed method. For an indefinite pti this does not hold in general.
§43- admissibility OF INDEFINITE TERMS l6l
For certain indefinite ptj we are sometimes able to find a synony-
mous definite^ ptj and by this means a method of resolution. But
this is not possible in the majority of cases.
Examples: We can represent the concept ‘prime number* by an
indefinite pt ‘ Primj* as 'well as by a synonymous definite pr ‘Primj*.
For example, we may define as follows (compare D ii, p. 6o):
‘Primi(x)= [^(jc = 0)« — (x= i). (u) ((m= i) v
iu = x)v^r\h(x,u))y;
and in the same way for *Prim2*, but with the restricted operator
*(u)x' instead of ‘ (u) *. Then ‘ Prim^ = Primj * is demonstrable ; and
thus the two pr are synonymous. On the other hand, for the in-
definite pr ‘Bewbll’ defined in II (where ‘Bewbll(a)’ means, in
syntactical interpretation: “The SNsentence a is demonstrable in
II**; see p. 75), no synonymous definite pr is known; and there is
reason to suppose (although so far it has not been proved) that no
pr of this kind exist. (The discovery of such a pr would mean ihe
discovery of a general method of resolution for II, and thus also for
classical mathematics.)
The lack of a method of resolution for indefinite terms has in-
duced many logicians to reject these terms altogether, as meaning-
less (e.g. Poincare, Brouwer, Wittgenstein, and Kaufmann). Let
us consider as examples two indefinite ^pr}, ‘Pi* and ‘Pg* (in II,
for example), which, by means of a definite ^ptf, ‘Q*, may be de-
fined in the following manner;
PiW=( 3 >’)(QC^,>')) (I)
p,(;f)=( 3 <)(Q(*,:v)) (2)
The logicians referred to argue roughly as follows: the question
whether, for instance, ‘ P^ (5) * (or ‘ P2 (5) * ) is true or not, is meaning-
less, inasmuch as we know of no method by which the answer may
be sought, and the meaning of a term consists solely in the method
of determination of its applicability or non-applicability. To this it
may be replied : it is true that we know of no method of searching
for the answer, but we do know what form the discovery of the
answer would take — that is to say, we know under what conditions
we should say that the answer had been found. This would be the
case, for example, if we discovered a proof of which the last sen-
tence was ‘ Pi (5) * ; and the question whether a given series of sen-
tences is a proof of this kind or not is a definite question. Thus
there exists the possibility of the discovery of an answer, and there
appears to be no cogent reason for rejecting the question.
SL
12
i 62
PART IV. GENERAL SYNTAX
Some logicians take the view that a question of this nature
is meaningless to begin with but becomes significant as soon
as an answer is discovered. We regard such an approach as par-
ticularly inexpedient. It leads to our considering, e.g. ‘Pi (5)?*
as a significant question, and ‘ (6) ? * as a meaningless one, or as
meaningless to-day and possibly significant to-morrow. This pro-
cedure is not, however, to be confused with the unquestionably use-
ful and universally applied method by which previously established
syntactical rules are altered as soon as certain fresh discoveries
have been made (for instance, concerning the mutual dependence
of primitive sentences, contradictions, etc.). As opposed to this,
in the former method reference to historical events is included
amongst the syntactical rules (concerning significance and non-
significance).
Sometimes, in the case of the rejection of indefinite pr, a further
distinction is made between the occurrence of an existential
operator and that of a universal operator. The reason advanced for
this differentiation is as follows: while for the proof of ‘ Pi (5) ’ the
discovery of a single number possessing the property designated
by ‘ Q (5, jv) ’ is sufficient, for the proof of ‘ Pg (5) * it must be shown
that every number possesses this property. There is, however, no
essential difference between these two cases. The discovery of a
number which has a particular, definite property, and the dis-
covery of a proof of a given sentence — that is to say, the discovery
of a sentence-series which has a particular, definite property — are
essentially similar operations; in both cases, it is a question of
discovering an element having a given definite property in a de-
numerable class (that is to say, in an infinite series progressing in
accordance with a given law). .
§ 44. On the Admissibility of
Impredicative Terms
Some logicians, while not rejecting all indefinite terms, reject a
number of them, namely, the so-called impredicative terms (e.g.
Russell in his so-called vicious-circle principle. See [Princ. Math.]
Vol. I, p. 37, and Fraenkcl [Mengenlehre]^ pp. 247 ff.). A thing is
usually called impredicative (in the material mode of speech) when
§ 44 - admissibility OF IMPREDICATIVE TERMS 163
it is defined (or can only be defined) with the help of a totality to
which it itself belongs. This means (translated into the formal
mode of speech) that a defined symbol Qi is called impredicative
when an unrestricted operator with a variable to whose range of
values Qj belongs, occurs in its chain of definitions. Example [(3)
serves only as an abbreviation];
M(F,*)= [(F(7).(y)[F(3;)DF(3-i)])3FW] (3)
P,(x) = (F)[M(F,x)] (4)
[‘^3(0)* means: “c possesses all the hereditary properties of 7.**]
As opposed to ‘Pj* and ‘Pg* (Examples in § 43), ‘P3* is not only
indefinite but impredicative as well, since it is of the same type
as ‘ F \ Now, against the admissibility of such a term, the following
objection is usually advanced. Assume that a concrete case is to
be decided, such as ‘ P3 (5) *, i.e. ‘ (F) [M (F, 5)] *. For this purpose
it must be determined whether every property has the relation M
tc 5; it must also be known, it is said, amongst other things,
whether this is true for P3. that is to say, whether ‘ M (P3, 5) * is
true. But this, according to (3), is equivalent in meaning to
‘ (P3 (7) • . . . ) 3 P3 (5) *. In order to find out the truth-value of this
implication, the values of both members must be established, and
hence also that of ‘ P3 (5) *. In short, in order to determine whether
‘P3(5)’ is true, a series of other questions must be answered,
amongst them whether ' P3 (5) ’ is true. This is said to be an obvious
circle ; therefore ‘ P3 (5) ’ is meaningless and consequently ‘ P3 ' also.
This form of argument seems, however, to be beside the point
(Carnap [Logiztsmus]) : in order to demonstrate the truth of a uni-
versal sentence, it is not necessary to prove the sentences which
result from it by the substitution of constants ; rather, the truth of
the universal sentence is established by a proof of that sentence
itself. The demonstration of all individual cases is impossible from
the start, because of their infinite number, and if such a test were
necessary', all universal sentences and all indefinite pr (not only the
impredicative ones) would be irresoluble and therefore (by that
argument) meaningless. As opposed to this, in the first place, the
construction of the proof is a finite operation ; and in the second
place, the possibility of the proof is quite independent of whether
the defined symbol occurs amongst the constant values of the
variable in question. In our example, ‘ M (Ps, 5) * can be resolved
164 PART IV. GENERAL SYNTAX
before we resolve ‘ P3 (5) * — for ‘ ^ M (P3, 5) ’ can easily be proved.
For the purposes of abbreviation, we define as follows :
'F,(x) = {x^6y.
Then first
‘~[(P4(7)*(>')[p«0')3P.(y)])3P4(s)]’
is demonstrable ; and next, from this,
•~M(P.,5 )’, ‘~(F)[M(F,s)]’.
and consequently ‘ P3 (5) * ; and similarly for every 33 from ‘ 0 * to
‘6' in place of ‘5*. Further, ‘P3(8)’ is easily demonstrable, and
similarly for every 33 from ‘ 7 * onwards.
In general, since there are sentences with unrestricted operators
which are demonstrable, there is always the possibility of coming
to q decision as to whether or not a certain indefinite or impredi-
cative term is applicable in a particular individual case, even
though we may not always have a method at hand for arriving at
this decision. Hence such terms are justified even from the stand-
point which makes the admissibility of any term dependent on the
possibility of a decision in every individual case. [Incidentally, in
my opinion, this condition is too narrow, and its necessity is not
convincingly established. ]
The proper way of framing the question is not “Are indefinite
(or impredicative) symbols admissible?” for, since there are no
morals in logic (see § 17), what meaning can * admissible’ have
here ? The problem can only be expressed in this way : “ How shall
we construct a particular language? Shall we admit symbols of
this kind or not ? And what are the consequences of either pro-
cedure ? ” It is therefore a question of choosing a form of language
— that is, of the establishment of rules of syntax and of the in-
vestigation of the consequences of these. Here, there are two
principal points to be considered : first of all, we have to decide
whether or not unrestricted operators are to be admitted, and
second, whether or not universal predicate-variables are to be
admitted for the different types. We will call pj universal when
all the constants of the type of p^ belong to the range of values of
Pi (that is to say, can be substituted for p^). In II all p are uni-
versal ; for instance, for a ^p^ any ^pr^ may be substituted. On the
other hand, in [Princ. Math.] the type (0), by the branched
rule of types, is divided again into sub-types, in such a way that
§ 44 - admissibility OF IMPREDICATIVE TERMS 165
for a particular p only the pr of a particular sub-type may be
substituted. — i. If the first point is decided in the negative and
unrestricted operators are excluded (as, for instance, in our
Language I), then all the indefinite and consequently all the im-
predicative symbols a*re excluded. If, however, we admit the un-
restricted operators, then the definiens of an indefinite definition
(compare Examples (i) to (4)) is in accordance with the rules of
syntax; but then it is natural to admit the definiendum as an
abbreviation for the definiens. — 2. The impredicative definitions
of pr of any types whatsoever can be excluded by deciding the
second point in the negative, and so not admitting universal
variables for these types. [In this way Russell rejects all universal
p, and Kaufmann all p in general.] If, however, we admit uni-
versal p and, moreover, admit them also in operators, then* the
definiens of an impredicative definition (compare Example (4)) is
in accordance with the rules of syntax. But then, again, it is
natural to admit the definiendum as an abbreviation for the
definiens. In any case, the material reasons so far brought for-
ward for the rejection either of indefinite or of impredicative terms
are not sound. We are at liberty to admit or reject such definitions
without giving any reason. But if we wish to justify either pro-
cedure, we must first exhibit its formal consequences.
§45. Indefinite Terms in Syntax
Our attitude towards the question of indefinite terms conforms
to the principle of tolerance; in constructing a language we can
either exclude such terms (as we have done in Language I) or ad-
mit them (as in Language II). It is a matter to be decided by con-
vention. If we admit indefinite terms, then strict attention must
be paid to the distinction between them and the definite terms;
especially when it is a question of resolubility. Now this holds
equally for the terms of syntax. If we use a definite language in the
formalization of a syntax (e.g. Language I in our formal construc-
tion), then only definite syntactical terms may be defined. Some
important terms of the syntax of transformations are, however,
indefinite (in general) ; as, for instance, ‘ derivable ’, ‘ demonstrable *,
and a fortiori ‘analytic’, ‘contradictory*, ‘synthetic*, ‘conse-
quence’, ‘content*, and so on. If we wish to introduce these
i66
PART IV. GENERAL SYNTAX
terms also, we must employ an indefinite syntax-language (such as
Language II).
In connection with the use of indefinite syntactical terms in the
construction of a particular language, we must above all differ-
entiate the formation and the transformation rules. The task
of the fonnation rules is the construction of the definition of
‘sentence*. This is frequently effected by defining a term ‘ele-
mentary sentence*, and determining several operations for the
formation of sentences. An expression is then called a sentence
when it can be constructed from elementary sentences by means
of a finite application of sentence-forming operations. Usually
the rules are so qualified that not only the terms ‘elementary
sentence* and ‘sentence-forming operation* but also the term
‘ sentence * is definite. In this case it can always be decided whether
a particular expression is a sentence or not. Although the adoption
of an indefinite term ‘sentence* is not inadmissible, it would in
most cases be inexpedient.
Examples of ‘ sentence * as an indefinite term : (i) Heyting [Math, i]
p. 5 ; the definition of ‘sentence* (there ‘expression*) is by rules 5.3
and 5.32 dependent upon the indefinite term ‘demonstrable’ (there
‘correct*), and is thus itself indefinite. (2) Diirr [Leihfiiz^ p. 87;
whether a certain combination of two sentences (‘ general value * and
‘principal value of the remainder*) is a sentence or not (there
‘significant* or ‘meaningless*) depends on the truth-values of the
two sentences; here therefore the term ‘sentence* is not only not
logically definite, but is moreover descriptive (i.e. dependent on
synthetic sentences). — If, in a language (e.g. in Peano), conditioned
definitions are admitted where "JIi is the de-
finiendum), then the term ‘sentence* is in general not logically
definite. An indefinite term ‘sentence* would perhaps be least open
to objection if it referred back to definite terms, ‘elementary sen-
tence* and ‘sentence-forming operation*. Von Neumann {[Be-
weisth.'\ p. 7) holds that the definiteness of the term ‘sentence* is
indispensable; otherwise the system is “incomprehensible and
useless**.
The principal terms concerning transformations, namely ‘de-
rivable* and ‘demonstrable*, are indefinite in the case of most
languages ; they are only definite in the case of very simple systems,
for instance in that of the sentential calculus. Nevertheless, we
can formulate the rules of transformation definitely, if, as is usually
done, we do not define those terms directly but proceed from the
§45- indefinite terms in syntax 167
definition of the definite terms ‘ directly derivable * (usually formu-
lated by means of rules of inference) and ‘primitive sentence*.
[Here ‘primitive sentence* can be represented as “directly de-
rivable from the null series of premisses *’ ; the definitions can be
taken as primitive sentences of a particular form. ] ‘ Derivable * is
determined by means of a finite chain of the relation ‘directly
derivable*; ‘demonstrable* is defined as “derivable from the null
series of premisses**. With the term ‘ consequence* (which has not
been defined in the languages in use hitherto), it is another matter.
Here the rules are indefinite even if they first define, not ‘ conse-
quence *, but only ‘ direct consequence * (as, for instance, those for
Language I in § 14).
B. THE SYNTAX OF ANY LANGUAGE
(a) GENERAL CONSIDERATIONS
§ 46. Formation Rules
In this section we shall attempt to construct a syntax for lan-
guages in general^ that is to say, a system of definitions of syntactical
terms which are so comprehensive as to be applicable to any
language whatsoever. [We have, it is true, had chiefly in mind as
examples languages similar in their principal features to the usual
symbolic languages, and, in many cases, the choice of the definitions
has been influenced by this fact. Nevertheless, the terms defined
are also applicable to languages of quite different kinds.]
The outline of a general syntax which follows is to be regarded as
no more than a first attempt. The definitions framed will certainly
need improvement and completion in many respects ; and, above all,
the connections between the concepts will have to be more closely
investigated (that is to say, further syntactical theorems will have to
be proved). As yet there have been very few attempts at a general
syntactical investigation; the most important are Tarski’s [Methodo-
logies and Ajdukiewicz*s [SpracheS-
By a language we mean here in general any sort of calculus,
that is to say, a system of formation and transformation rules con-
cerning what are called expressions y i.e. finite, ordered series of cle-
i68
PART IV. GENERAL SYNTAX
ments of any kind, namely, what are called symbols (compare §§ i
and 2). In pure syntax, only syntactical properties of expressions,
in other words, those that are dependent only upon the kind and
order of the symbols of the expression, are dealt with.
As opposed both to the symbolic languages of logistics and to the
strictly scientific languages, the common word-languages contain
also sentences whose logical character (for example, logical validity
or being the logical consequence of another particular sentence, etc.)
depends not only upon their syntactical structure but also upon extra-
syntactical circumstances. For instance, in the English language, the
logical character of the sentences ‘yes* and ‘no*, and of sentences
which contain words like ‘he*, ‘this* (in the sense of “the afore-
mentioned **) and so on, is also dependent upon what sentences have
preceded them in the same context (treatise, speech, conversation,
etc.). In the case of sentences in which words like ‘ I*, ‘ you *, ‘ here *,
‘noTv*, ‘to-day*, ‘yesterday*, ‘this* (in the sense of “the one pre-
sent ’*) and so forth occur, the logical character is not only dependent
upon the preceding sentences, but also upon the extra-linguistic
situation— rnamely, upon the spatio-temporal position of the
speaker.
In what follows, we shall deal only with languages which contain
no expressions dependent upon extra-linguistic factors. The logical
character of all the sentences of these languages is then invariant in
relation to spatio-temporal displacements ; two sentences of the same
wording will have the same character independently of where, when,
or by whom they are spoken. In the case of sentences having extra-
syntactical dependence, this invariance can be attained by means of
the addition of person-, place-, and time-designations.
In the treatment of Languages I and II we introduced the term
‘ conseqtience ’ only at a late stage. From the systematic standpoint^
however^ it is the beginning of all syntax. If for any language the
term 'consequence' is established^ then everything that is to be said
concerning the logical connections within this language is thereby
determined. In the following discussion we assume that the trans-
formation rules of any language S, i.e. the definition of the term
‘direct consequence in S*, are given. [For the sake of brevity in
the case of syntactical terms, we usually leave out the specification
‘in S* or ‘of S’.] We shall, then, show how the most important
syntactical concepts can be defined by means of the term ‘ direct conse-
quence *. In this process it will become clear that the transforma-
tion rules determine, not only concepts, such as ‘ valid ’ and ‘ con-
tra-valid’, but also the distinction between logical and descriptive
symbols^ between variables and constants^ and further, between logical
§46. FORMATION RULES
169
and extra-logical {physical) transformation rules, from which the
difference between ‘valid* and ‘analytic* arises; also that the dif-
ferent kinds of operators and the various sentential connections can
be characterized, and the existence of an arithmetic and an in-
finitesimal calculus in*S can be determined.
As syntactical Gothic symbols, we use (as previously) ‘a’ for
symbols, ‘?t* for (finite) expressions, ‘il* for (finite or infinite)
classes of expressions (for the most part, of sentences). All further
Gothic symbols in the general syntax (even those used previously
in I and II) are defined in what follows. We say of an expression
that it has the form %
r%]
w
when it results from?Ii by the replace-
ment at some place in of a partial expression by 3I3. (On the
difference between replacement and substitution, see pp. 36 1)
We restrict ourselves to finite expressions only because, up to
now, there has been no particular reason for dealing with infinite
expressions. There is no fundamental objection to the introduction
of infinite expressions and sentences. The treatment of them in an
arithmetized syntax is quite possible. While a finite expression is
represented by a series of numbers which can be replaced by a single
series-number, an infinite expression would have to be represented
by an infinite series of numbers or a real number. Such a series is
expressed by means of a (definite or indefinite) functor. According
to what was said previously (§ 39) we can speak not only of infinite
expressions which are systematically constructed, but also of infinite
expressions which are not determined by any mathematical law.
An fui corresponds to the former, an fu^ to the latter.
We will assume the definition of * direct consequence' to be stated
in the following form: “ 9 Ii is called a direct consequence of 5 li in
S if: (i) and every expression of 5 ti has one of the following
forms : . . . , and (2) and fulfil one of the following conditions :
....** The definition thus contains under (i) the formation rules
and under (2) the transformation rules of S. Now we call 9I2 a
sentence (S) if ^ has one of the forms under (i). Those a that are
S are called sentential symbols (fa).
and 9I2 (an a is also an ?l) are said to be syntactically related
when there exists an Si such that ?Ii occurs in Sj and Si
a sentence. Two related expressions ' 2 ti and %2 called isogenous
if for any Si, Si [”^1 sentences. A class fti of
170
PART IV. GENERAL SYNTAX
expressions is called a genus if every two expressions of are
isogenous, and no expression of is isogenous with an expression
which does not belong to [Relatedness is a similarity (on
these and the following terms see Carnap [Logistik] p. 48) ; further,
isogeneity is transitive, and therefore an equality; the genera are
the abstractive classes with respect to isogeneity ; hence different
genera have no members in common. ] The sub-class of a genus
of expressions which contains all the symbols and only the symbols
of this genus is called a symbolic genus. Every of S belongs to
exactly one genus; if the genus of ?ti is so that is not
isogenous with some unequal then is called isolated. Two
expressional genera or two symbolic genera are called related when
at least one expression of the one is related to one of the other ; in
this case every expression of the one is related to every expression
of the other.
In what follows, definitions of further syntactical formation
terms will result from the transformation terms.
Examples: In I and II every 3 is isolated ; for (Zi) is not
a sentence. In Hilbert’s symbolism also, every 3 is isolated; here,
namely, (3i) (pri(3i))Q^J for unequal 3^ and 32 is not a sentence.
In I and II all constant 33 together form a genus. On the other hand,
in I and II 3^ and nu, for example, are related but not isogenous,
since in an operator 3i cannot be replaced by nu.
The or 5 ** of any type ^ in II are to be divided into two related
genera; that of the p (or f) of t and that of the remaining "ipr (or gu,
respectively). Thus the pr (or fu) of t are to be divided into two
related symbolic genera: that of the p (or f) of t and that of the
constant pr (or fu, respectively) of L
§ 47. Transformation Rules ; d-Terms
We will now assume that the transformation rules of S which
have been given in one way or another are converted into the form
previously indicated of a definition of * direct consequence in S ’.
It makes no difference in what tenninology the rules were origin-
ally stated ; all that is necessary is that it be clear to what forms of
expressions the rules are in general applicable (which gives us the
definition of ‘ sentence ’) and under what conditions a transforma-
tion or inference is permitted (which gives us the definition of
direct consequence ’). -
§ 47* transformation rules ; d-terms i 7 i
For instance, instead of ‘ direct consequence *, we frequently have
the terms ‘derivable*, ‘deducible*, ‘inferable*, ‘results from*, ‘may
be concluded (inferred, derived...) from*, etc.; and, instead of
‘direct consequence of the null class*, it is customary to find
‘primitive sentence*, ‘axiom*, ‘true*, ‘correct*, ‘demonstrable*,
‘logically valid*, etc. We shall assume that even those rules con-
cerning symbols of S, that are usually designated as definitions, are
included in the rules concerning ‘ direct consequence * (for instance,
as primitive sentences or rules of inference of a special kind); the
definitions can either be finite in number and stated singly, or un-
limited in number and established by means of a general law (as,
for example, in I and II).
The second part of the definition of ‘direct consequence* con-
sists of a series of rules of the following form: “ Sj is a direct con-
sequence of the sentence-class fti if (but not only if) and Sii
have such and such syntactical properties.** We will extend .this
series by means of the following rule (which sometimes already
belongs to the original series) : “ Si is always a direct consequence
of {Si}.** We call the rules of the whole series ru/es of consequence^
or, briefly, c-rules. Those in which the properties stipulated for Si
and i^i are definite we call rules of derivation, or, briefly, d-ruJes.
Sj is called directly derivable from 5^i if 5li and Si satisfy one of the
d-rules. Si is called a primitive sentence if Si is directly derivable
from the null class. A finite series of sentences is called a deriva-
tion with the premiss class SKi if every sentence of the series either
belongs to 5ti or is directly derivable from a class ftg* ^be sentences
of which precede it in the series. A derivation with a null premiss-
class is called a proof. Si is called derivable from (or a d-conse-
quence of) the sentential-class 5ti if Si is the last sentence of a
derivation v\ ith the premiss-class fti. Si (or 5ti) is called demon-
strable (or d-valid) if Si (or every sentence of ftj, respectively) is
derivable from the null class and is therefore the last sentence of a
proof. Si (or fti) is called refutable (or d-contravalid) if every
sentence of S is derivable from {Si} (or 5^^, respectively). Sj (or
ill) is called resoluble (or d -determinate) if Si (or Rj, respectively)
is either demonstrable or refutable; otherwise irresoluble (or
d-indeterminate).
Let be the largest class of symbols in S having the following
properties. The symbols of fti can be arranged (not necessarily uni-
vocally) in a series. If Qj belongs to fti, then there is by the d-rules
a definite direction for construction (in an arithmetized syntax,
172
PART IV. GENERAL SYNTAX
that means a definite syntactical functor), according to which, for
every sentence in which Oi occurs, a sentence 02
structed such that 02 does not contain Ui, but only symbols which
either do not belong to or which precede in that series, and
such that 01 and 02 are derivable from one another. We call such
a direction a definition of Qi, and the transformation of 0 ^ into 02
the elimination of Di- We call the symbols of defined, the
others undefined.
We divide the syntactical terms into d-terms and c-terms,
according to whether their definition refers only to the d-rules (as
for instance in the preceding definitions) or to c-rules in general.
§48. c-Terms
We shall now define a number of c-terms, beginning with ‘ con-
sequence one of the most important syntactical terms. In what
follows the R are always sentential classes. 0^ is called a conse-
quence of 5^1, if 01 belongs to every sentential class R^ satisfying
the following two conditions: i. 5 li is a sub-class of R^ \ 2. Every
sentence which is a direct consequence of a sub-class of R^ belongs
to Ri* 5^2 is called a consequence-class of if every sentence
of 5^2 is ^ consequence of 5 lj. If d-rules only are given, then the
terms ‘derivable* and ‘consequence* coincide; and if the term
‘ direct consequence ’ already possesses a certain kind of transitivity
then it coincides with ‘ consequence
What has previously been said in the case of Language I holds
in general for the fundamental difference between ‘ derivable in S *
and ‘ consequence in S' (see pp. 38!.), and analogously for every pair
which consists of a d-term and its correlative c-term ; compare the
second and third columns in the survey on p. 183.
In almost all known systemsy vfily definite rules of transformation
are statedy that is to say, only d-rulcs. But we have already seen
that it is possible to use also indefinite syntactical terms (§45). We
shall therefore admit the possibility of laying down indefinite
transformation rules and of introducing the c-terms which are
based upon these. In dealing with the syntax of Languages I and
• (Note, 1935.) The above definition of ‘consequence* is a cor-
rection of the German original, the need for which was pointed out
to me by Dr. Tarski.
173
§48. C-TERMS
TI we have come to recognize both the importance and the fer-
tility of c-terms (such as ‘consequence*, ‘analytic*, ‘content*,
etc.). One important advantage of the c-terms over the d-terms
consists in the fact that with their help the complete division of Si
into analytic and cor>tradictory is possible, whereas the corre-
sponding classification of Si into demonstrable and refutable is
incomplete.
Only d-rules arc given in the systems of Russell [Princ. Math.'],
Hilbert [Logik], vo*: Neumann [Beweisth.], Godel [Unentscheidbare],
Tarski [ Wider sprue lufr .] .
Hilbert [GrundL 1931] [Tertium] recently stated a rule of trans-
formation which (in our terminology) runs approximately as follows :
“If Si contains exactly one free variable 3i, and if every sentence of
the form Si is demonstrable, then (Si) (Si) may be laid down as
a primitive sentence.** Hilbert calls this rule a “new finite rule of
inference**. What is to be understood by ‘finite* is not precisely
stated; according to indications given by Bernays [Pkilosophie]
P* 343 » it means about what we mean by ‘definite*. The rule is,
however, obviously indefinite. Its formulation was presumably
motivated by the incompleteness, indicated above, of all arithmetics
which are restricted to d-rules. The rule given, however, which
refers only to numerical variables 3, is not sufficient to secure a com-
plete classification.
Herbrand [Non~contrad.] p. 5 makes use of Hilbert*s rule, but
with certain restrictions; Si and the definitions of the fu which
occur in Si must not contain any operators.
Tarski discusses Hilbert’s rule (“Rule of infinite induction**
[Wider spruchsfr.] p. 1 1 1) — he himself had previously (1927) laid down
a similar one — and rightly attributes to it an “infinitist character**.
In his opinion : “ it cannot easily be harmonized with the interpreta-
tion of the deductive method that has been accepted up to the
present**; and this is correct in so far as this rule differs funda-
mentally from the d-rules which have hitherto been exclusively used.
In my opinion, however, there is nothing to prevent the practical
application of such a rule.
In Language I, DC i refers back to the definite rules PS i-ii
and RI 1-3 ; DC 2 is indefinite.
1 is called valid if Sii is a class of consequences of the null class
(and hence of every class). [We do not use the term ‘analytic’
here because we wish to leave open the possibility that S contains
not only logical rules of transformation (as do Languages I and II)
but also physical rules such as natural laws (see § 51). In relation
to languages like I and II, the terms ‘ valid ’ and ‘ analytic ’ coin-
m
PART IV. GENERAL SYNTAX
cide.] Ri is called contravalid if every sentence is a consequence
of 5 li. fti is called determinate if Ri is either valid or contravalid ;
otherwise indeterminate. In a word-language it is convenient in
many cases to use the same term for properties both of sentences
and of classes of sentences. We shall call a sentence Si valid (or
contravalid, determinate, or indeterminate) if {Si} is valid (or con-
travalid and so on, respectively). And we shall proceed in the same
way with the terms which are to be defined later.
Theorem 48.1. Let 5^2 ^ consequence-class of Ri ; if Ri is
valid, R2 is also valid ; if R2 is contravalid, so also is R^.
Theorem 48.2. Let S2 be a consequence of Si ; if 0 i is valid,
S2 is also valid ; if S2 is contravalid, so also is Si.
Theorem 48.3. If every sentence of Ai is valid, R^ is also valid ;
and conversely.
Theorem 48.4* If at least one sentence of R^ is contravalid, then
Ri is contravalid; the converse is not universally true.
Two or more sentences are called Incompatible (or d-incom-
patible) with one another if their class is contravalid (or refutable,
respectively); otherwise they are called compatible (or d-com-
patible). Two or more sentential classes are called incompatible
(or d-incompatible) with one. another if their sum is contravalid
(or refutable, respectively); otherwise they are called compatible
(or d-compatible).
R2 is called dependent upon R^ if R2 is a consequence-class of
5 ti, or is incompatible with Ri; otherwise it is called independent
of ftj. ^2 'S called d-de'pendent upon R^ if either every sentence of
R2 is derivable from fti, or 5^2 is d-incompatible with R ^ ; other-
wise it is called d-independent of R^, (The definitions are ana-
logous for Sj and 02*)
Theorem 48.5. If is dependent (or d -dependent) upon the
null class, then Ri is determinate (or resoluble, respectively) ; and
conversely.
We say that there is (mutual) independence within R^ if every two
sentences of Ri are independent of one another. And we say that
there is complete independence within 5 li if every proper non-null
sub-class of 5^1 is independent of its complementary class in
Theorem 48.6. If Ri is not contravalid and is not a consequence-
class of a proper sub-class, then there is complete independence
within ill 9 conversely.
§48. C-TERMS 175
ill is called complete (or d-complete) if every il (and conse-
quently every S of S) is dependent (or d-dependent, respectively)
upon ill ’ otherwise it is called incomplete.
Theorem 48.7. If ilg is complete and is a consequence-class of
ill, then ill ^^^o is complete.
Theorem 48.8. If the sentential null class is complete (or d-
complete) in S, then every il in S is complete (or d-complete,
respectively).
The arrows in the table on p. 183 indicate the dependence be-
tween the defined d- and c-concepts. Although the d-method is the
fundamental method and the d-terms have the simpler definitions,
yet the c-terms are the more important from the standpoint of
certain general considerations. They are more closely connected
with the material interpretation of language; and this is shown
formally by the fact that simpler relations obtain among them.
In what follows we shall be dealing principally with the c-terms,
and shall only state the corresponding d-terms occasionally (if no
special term is given, one is constructed from the c-term by pre-
fixing a ‘ d- ’).
§ 49. Content
By the content of 5 li (or of Sj^; cf. p. 174) in S, we understand
the class of the non-valid sentences which are consequences of 5li
(or 0 ]^, respectively). This definition is analogous to the previous
definitions for Language I (p. 42) and Language II (p. 120); it
must here be noted that in Languages I and II ‘valid* coincides
with ‘ analytic *.
Other possibilities of definition. Instead of the class of the non-
valid consequences, one might perhaps designate as ‘content’ the
class of all consequences. As opposed to this, our definition has the
advantage that by it the analytic sentences in pure L-languages (see
below) such as I and II have the null content. Again, it might be
possible to take as ‘content’ the class of all indeterminate conse-
quences, or even the class of all non-contra valid consequences. Let
S be a non-descriptive language (such as a mathematical calculus).
Then, in S there are no indeterminate (or synthetic) senteAces. In
this case, on the basis of our definition, the analytic sentences are
equipollent, and similarly the contradictory sentences; but there is
not equipollence between the two. On the basis of either of the above
definitions, on the other hand, all sentences would be equipollent,
PART IV. GENERAL SYNTAX
176
though they differ essentially from one another in that only analytic
sentences are consequences of an analytic sentence, but all sentences
are consequences of a contradictory sentence. Ajdukiewicz gives
a formal definition of ‘sense’ which is worthy of note. It differs
considerably from our definition of ‘ content *, for, according to it,
the term ‘ equivalence of sense * is very much narrower than our term
‘ equipollence
and 5^2 are called equipollent when their contents coincide.
If the content of 5^2 is a proper sub-class of the content of then
5^2 is called poorer in content than 5 ti, and R^ richer in content than
R^. We say that R^ has the null content if the content of fti is
empty, i.e. the null class. We say that R^ has the total content if the
content of R^ is the class of all non-valid sentences. Two or more
classes are said to have exclusive contents if their contents have no
member in common. All these terms are also applied to sentences
(see p. 174). We say that a mutual exclusiveness in content subsists
in if every two sentences of R^ have exclusive contents.
TheoreiSa 49.i« If 5^2 is a consequence-class of Ai, then the
content of A2 is contained in that of Ai*, and conversely. In the
transition to a consequence^ an increase in the content never occurs.
It is in this that the so-called tautological character of the conse-
quence-relation consists.
Theorem 49.2. If A^ and A2 are consequence-classes of one
another, then they are equipollent and conversely.
Theorem 49.3. If A2 is a consequence-class of Ai, but Ai not a
consequence-class of ilgi i® ^^er in content than A2 ; and
conversely.
Theorem 49.4. If Ai is valid^ then Ai has the null content ; and
conversely.
Theorem 49«5. If Ai is contravalid, then Ai has the total content ;
and conversely.
Theorems i to 5 hold likewise for Si and S2.
Ai is called perfect if the content of Ai is contained in Ai.
According to this, every content is perfect. The product of two
perfect classes is also perfect ; but this is in general not true for the
sum.
is said to be replaceable by ^2 if i® always equipollent to
^1 and ^ are called synonymous (with one another) if
they are mutually replaceable. Only expressions of the same genus
§ 49 * CONTENT 177
can be synonymous. [If is replaceable by Sla, it is usually also
synonymous with ]
is called a principal expression if is not empty and there
exists an expression which is related to, but not synonymous with,
We count as principal symbols^ first, every symbol which is a
principal expression, and, second, symbols of certain kinds which
will be described later on (e.g. 93, u, pr, uf , 33) ; the rest of the
symbols are called subsidiary symbols. [Example: The principal
symbols of Language II are the fa, 33, pr, fu, ucrfn, and
‘ *'*» ‘3* (by the definitions of general syntax, is a of,
‘ = * a pr, ‘ ’ a 3fu ; the null expression is related to ‘ 3 * but is not
synonymous with it). The remaining symbols are subsidiary
symbols, namely, brackets, commas, and ‘ K* (because in II there
are no numerical operators other than the K-operators). ]
§ 50. Logical and Descriptive Expressions ;
Sub-Languages
If a material interpretation is given for a language S, then the
symbols, expressions, and sentences of S may be divided into
logical and descriptive, i.e. those which have a purely logical, 01
mathematical, meaning and those which designate something
extra-logical — such as empirical objects, properties, and so forth.
This classification is not only inexact but also non-formal, and
thus is not applicable in syntax. But if we reflect that all the con-
nections between logico-mathematical terms are independent of
extra-linguistic factors, such as, for instance, empirical observa-
tions, and that they must be solely and completely determined by
the transformation rules of the language, we find the formally
expressible distinguishing peculiarity of logical symbols and ex-
pressions to consist in the fact that each sentence constructed
solely from them is determinate. This leads to the construction of
the following definition. [1 he definition must refer not only to
symbols but to expressions as well ; for it is possible for Oi in S to
be logical in certain contexts and descriptive in others.]
Let Ri be the product of all expressional classes Ri of S,
which fulfil the following four conditions. [In the majority of
the usual language -systems, there exists only one class of the
kind 51 , ; this is then 5 ^^.] i. If 9 li belongs to 5 l„ then 9 Ii is not
SL
13
178 PART IV. GENERAL SYNTAX
empty and there exists a sentence which can be sub-divided into
partial expressions in such a way that all belong to and one of
them is 2. Every sentence which caa be thus sub-divided into
expressions of is determinate. 3. The expressions of are as
small as possible, that is to say, no expressio'a belongs to ft,- which
can be sub-divided into several expressions of ft<. 4. ft, is as com-
prehensive as possible, that is to say, it is not a proper sub-class of
a class which fulfils both (1) and (2). An expression is called logical
(9Ii) if it is capable of being sub-divided into expressions of ftj ;
otherwise it is called descriptive A language is called logical
if it contains only af, otherwise descriptive.
With a language which is used in practice — for instance, that of
a particular domain of science — it is usually quite clear whether a
certain symbol has a logico-mathematical or an extra-logical, say a
physical, meaning. In an unambiguous case of this kind, the
formal differentiation just given coincides with the usual one.
There are occasions, however, when a mere non-formal considera-
tion leaves it doubtful whether a symbol is of the one kind or the
other. In such a case, the formal criterion helps us to a clear de-
cision, which on closer examination will also be found to be
materially satisfactory.
Example: Is the metrical fundamental tensor * g^^ \ by means of
which the metrical structure of physical space is determined, a
mathematical or a physical term ? According to our formal criterion,
there are here two cases to be distinguished. Let Si and Sj be
physical languages, each of them containing not only mathematics
but also the physical laws as rules of transformation (this will be
examined more closely in § 51). In Si a homogeneous space may be
assumed : ‘ g^ * has the same value everywhere, and at every point
the measure of curvature is the same in all directions (in the simplest
case, 0 — Euclidean structure). In S2, on the other hand, the Ein-
steinian non-hpmogeneous space may be assumed: then *g^’ has
various values, depending upon the distribution of matter in space.
They are therefore — and this is an essential point for our differentia-
tion — not determined by a general law. ‘ * is thus a logical symbol
in Si and a descriptive symbol in S2. For the sentences which give
the values of this tensor for the various space-time points are in Si
all determinate ; and on the other hand, in S2 at least part of them
are indeterminate. At a first glance, it may appear strange that the
fundamental tensor should not have the same character in all
languages. But on closer examination we must admit that there is
here a fundamental difference between Si and S2. The metrical
calculations (for example, the calculation of a triangle from suitable
§ 50. LOGICAL AND DESCRIPTIVE EXPRESSIONS 179
determinations) are made in Si by means of mathematical rules
which, it is true, in some respects (for instance, in the choice of the
value of a fundamental constant such as the constant curvature of
space) are based on empirical observations (see § 82). But on the
other hand, for such calculations in Sj empirical data are regularly
required, namely, data concerning the dictribution of the values of
the fundamental tensor (or of the density) in the space-time domain
in question.
Theorem 50.1. Every logical sentence is determinate; every in-
determinate sentence is descriptive. With the given form of de-
finition for * logical * this follows directly. If ‘ logical expression ’
is defined in some other way (for instance, by the statement of the
logical primitive symbols, as in Languages I and II) then the de-
finitions of the terms ‘valid’ and ‘ contravalid ’ (which in I and II
coincide with ‘ analytic ’ and ‘ contradictory ’) must be so contriv.ed
that every Si is determinate.
Theorem 50.2. (a) If S is logical, then every ft in S is deter-
minate; and conversely, {b) If S is descriptive, then there is an
indeterminate ft in S ; and conversely.
S2 is called a sub-language of Sj if the following conditions hold :
I. every sentence of Sg is a sentence of S^; 2. if ftg is a conse-
quence-class of fti in Sg, then it is likewise a consequence-class of
ftj in Sj. Sg is called a conservative sub-language of when, in
addition : 3. if ftg is a consequence-class of ftj in Sj, and ftg and ft^
also belong to Sg, then ilg is also a consequence-class of ft^ in Sg.
If Sg is a sub-language of S^ but not S^ of Sg, then Sg is called a
proper sub-language of S^. By the logical sub-language of S, we
understand the conservative sub-language of S which results from
S by the elimination of all the descriptive sentences.
Let Sg be a sub -language of S3, and ftj and ftg sentential classes
of Sg. The table on p. 225 states under what conditions a syn-
tactical property of ft^, or a relation between ft^ and 5^2, which
obtains in Sg, obtains also in S3 (rubric 3) ; or conversely (rubric 5).
Thus, for example, we can see from the table that if ft^ is valid in
Sg, then it is also valid in S3 ; if ftj is valid in S3 and Sg is a con-
servative sub-language of S3, then ft^ is also valid in Sg.
Example: I is a proper conservative sub-language of II. Let I' be
the language which results from I if unrestricted operators with 5
are admitted; then I is a proper sub-language of I' although both
languages possess the same symbols.
i8o
PART IV. GENERAL SYNTAX
§ 51. Logical and Physical Rules
For Languages I and II we have set up only rules of trans-
formation that on a material interpretation can be represented
as having a logico-mathematical basis. The same is true of the
majority of symbolic languages which have hitherto been formu-
lated. We may, however, also construct a language with extra-
logical rules of transformation. The first thing which suggests itself
is to include amongst the primitive sentences the so-called laws of
nature, i.e. universal sentences of physics (‘ physics * is here to be
understood in the widest sense). It is possible to go even further
and include not only universal but also concrete sentences — such
as empirical observation-sentences. In the most extreme case we
may even so extend the transformation rules of S that every sen-
tence which is momentarily acknowledged (whether by a par-
ticular individual or by science in general) is valid in S. For
the sake of brevity, we shall call all the logico-mathematical
transformation rules of S logical or L-rw/w; and all the re-
mainder, physical or V-rules. Whether in the construction of a
language S we formulate only L-rules or include also P-rules, and,
if so, to what extent, is not a logico-philosophical problem, but a
matter of convention and hence, at most, a question of expedience.
If P-rules are stated, we may frequently be placed in the position
of having to alter the language ; and if we go so far as to adopt all
acknowledged sentences as valid, then we must be continuously
expanding it. But there are no fundamental objections to this.
If we do not include certain acknowledged sentences as valid in S,
this does not mean that they are excluded from S. They can still
appear in S as indeterminate premisses for the derivation of other
sentences (as for instance all the synthetic sentences of I and II).
Now how is the difference between h-rules and V-rules — which we
have here only indicated in an informal way — to be formally de-
fined} This difference, when related to primitive sentences, does
not coincide with the difference between logical and descriptive
sentences. An Si as a primitive sentence is always an L-rule ; but
an Sb as a primitive sentence need not be a P-rule. [Example: Let
‘ Q ’ be a ptb of Language I. Then, for example,
‘Q(j)d(~Q(3)3Q(s))’ (Si)
§51. LOGICAL AND PHYSICAL RULES l8l
is a descriptive primitive sentence of the kind PSI i. But Sj is
obviously true in a purely logical way, and we must arrange the
further definitions so that Si is counted amongst the L-rules and
is called, not P-valid, but analytic (L-valid). That Si is logically
true is shown formally <by the fact that every sentence which results
from Si when ‘Q’ is replaced .by any other pr is likewise a
primitive sentence of the kind PSI i.] The example makes it clear
that we must take the general replaceability of the as the de-
finitive characteristic of the L-rules.
Let S2 be a consequence of fti in S. Here three cases are to be
distinguished : i. Ai and S2 s^re logical. 2. Descriptive expressions
occur in 5li and in S2, but only as undefined symbols; here
two further cases are to be distinguished : 2 a. for any ^3 and S4
which are formed from fti (or Sg) the replacement of every
descriptive symbol of ilj (or S2 respectively) by an expression of
the same genus, and specifically of equal symbols by equal ex-
pressions, the following is true : S4 is a consequence of ft, ; 2 b. the
condition mentioned is not fulfilled for every ft, and S4. 3. In ftj
and S2 defined descriptive symbols also occur; let Aj and (So be
constructed from ft^ (or 62 respectively) by the elimination of every
defined descriptive symbol (including those which are newly intro-
duced as the result of an elimination) ; 3 a. the condition given in
2 a for fti and S2 is fulfilled for Ai and (S, '» 3 b, the said condition
is not fulfilled. In cases i, 2 a, 3 a, we call 0 , L-conse-
quence of ft^ ; in cases 2 3 6, we call Sg ^ P-conscquence of ft^.
Thus the formal distinction between L- and P-rules is achieved.
If S contains only L-niles (that is to say, if every consequence in
S is an L-consequence), we call S an L-language ; otherwise, a
P-language. By the L-sub -language of S we shall mean that sub-
language of S which has the same sentences as S but which has as
transformation rules only the L-rules of S.
Theorem 51.1. Every logical language is an L-language. The
converse is not always true.
The distinction between L- and P 4 anguages must not be confused
with that between logical and descriptive languages. The latter is
dependent upon the symbolic apparatus (although only, it is true,
upon a property of the symbolic apparatus which appears in the
transformation rules), the former on the kind of the transforma-
tion rules. Languages I and II are, for example, descriptive
i 82
PART IV. GENERAL SYNTAX
languages (they contain Ob, as is shown by the occurrence of in-
determinate, namely, synthetic sentences), but they are L-lan-
guages : every consequence-relation in them is an L-consequence ;
and only analytic sentences are valid in them. Similarly, the dif-
ference between the L-sub-language of 5 and the logical sub-
language of S is to be noted. For instance, if S is a descriptive
L-language (like I and II) then the L-sub-language of S is S itself,
but the logical sub-language of S is a proper sub-language.
§ 52. L-Terms ; ‘ Analytic ’ and ' Contradictory *
To the previously defined d- and c-terms we now add L-terms
(to wit, L-d-terms and L-c-terms). If in the L-sub-language of S,
has a particular (d- or c-) property, we attribute to it in S the
corresponding L-property. For instance, Si is called L-demon-
strable in S if Si is demonstrable in the L-sub-language of S.
5^2 is called the L-content of in S if 5^2 is content of fti in the
L-sub-language of S, and so on. Instead of ‘ L-valid ' L-contra-
valid*, and ‘ L-indeterminate we shall usually say 'analytic’,
'contradictory*, and 'synthetic*. In the table which follows
(p. 183), the correlative terms are placed on the same line. An arrow
between two terms shows that one may be inferred from the other.
[Example: If S2 is L-derivable from Si then it is also derivable
from Si ; and if derivable from Si, then also a consequence of Si-
Between an L-d- and an L-c-term, the inference always holds in
the same direction as between the correlative d- and c-terms.]
Here again the d- and L-d-terms are more fundamental for the
method of proof; on the other hand, the c- and L-c-terms are the
more important for many applications.
Since I and II are L-languages, in their case every syntactical
term coincides with the correlative L-term (for instance, ‘de-
monstrable * with ‘ L-demonstrable ‘ consequence * with ‘ L-conse-
quence *, ‘ valid * with ‘ analytic \ ‘ content * with ‘ L-content *, and so
on). The L-d- and L-c-terms which were previously defined for I
and II agree with ♦hose now defined, even where the earlier de-
finition has quite a different form (as for example in the case of
'analytic in II’).
Theorem 52.1. (a) Every analytic sentence is valid, (b) Every
valid logical sentence js analytic. — Regarding (^) ; Let Si be a valid
§52- L-TERMS ; ‘ ANALYTIC ’ AND ‘ CONTRADICTORY ’
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183
184 PART IV. GENERAL SYNTAX
Sf. Then Sj is a consequence of the null class, and hence an L-
consequence of it, and therefore analytic.
Theorem 52,2. {a) Every contradictory sentence is contravalid.
(A) Every contravalid logical sentence is contradictory. — Regarding
(6) : Let be a contravalid Sj. Then every sentence is a conse-
quence of Sj. Therefore, in the first place, every Sj, and in the
second place, in the case of all every Sd transformed according
to rule 2 a or 3 a (p. 181), is a consequence of Sj. Hence every sen-
tence is an L-consequence of Sj. Therefore, Sj is contradictory.
Theorem 52.3. Every logical sentence is L-determinate ; there
are no synthetic logical sentences. This follows from Theorems
50.1, 52.1 ft and 2^.
Theorem 52.4. If every sentence of is analytic, then 5 ^^ is
anal3rtic; and conyersely.
Theorem 52.5. If at least one sentence of is contradictor^'
then Ri is contradictory. If is logical, then the converse is also
true.
Theorem 52.6. Let ®2 ^ consequence of R^. {a) If R^ is
analytic, then Sg is also analytic, (ft) If S2 is contradictory, then
Ri is also contradictory.
Theorem 52.7. If Sj is an L-consequence of the sentential null
class (and therefore of every class), then ®i is analytic ; and con-
versely.
Theorem 52.8. If Rj is contradictory, then every sentence is an
L-consequence of R^ ; and conversely.
Theorem 52.9. The L-content of R^ is the class of the non-
analytic sentences that are L-consequences of R^.
The ordinary concept of the equivalence in sense ot two sentences
is ambiguous. We represent it by means of two different formal
terms, namely, equipollence and L-equipollence. Analogously, we
replace the ordinary concept of the equivalence in meaning of two
expressions by two different terms, synonymity and L-synonymity.
(Compare § 75 : Examples 6-9.)
The \^-terms are obtained by restriction to the L-rules of the lan-
guage. For some of these terms, we will define corresponding P-
terms. These are characterized by the fact that, for them, the P-rules
also are taken into account. In L-languages, they are empty. Sg
called a V-consequence of R^ if Sg is a consequence, but not an
L-consequence, of R^. R^ (or Sj) is P-valid if it is valid but not
§ 52. L-TERMS ; ‘ ANALYTIC * AND ‘ CONTRADICTORY * 1 85
analytic, (or Sj) is P-contravalid if it is contravalid but not
contradictory. and ^2 ^-equipollent if they are equipollent
but not L-equipollent. and % are '9 -synonymous if they are
synonymous but not L-synonymous. In what follows we shall
make very little use o^ the P-terms.
For a P-language we get the following classification of descriptive
sentences (for the Sj, see p. 210 ):
(d-terms :)
(P-terms :)
(L-terms :)
(c- terms :)
demonstrable irresoluble refutable
P-valid P-contra-
valid
(L-valid)
analytic
Y
valid
synthetic
indeterminate
(L-contravalid)
contradictory
-V- '
contravalid
For an L-language (such as I and II) the classification of the de-
scriptive sentences is simpler, since the c- and L-c-terms coincide :
(d-terms :)
(c- and L-
terms:)
demonstrable
A
irresoluble
X
refutable
A
-- I-,
Y >
1 1
1
1 1
A . . J
valid
analytic
Y
indeterminate
synthetic
contravalid
contradictory
Examples: Assume that S is a P-language with English words used
in their ordinary meaning. Let the most important physical laws be
stated as primitive sentences of S. Let Si be : ‘ this body a is of iron ’ ;
S2 : ‘ a is of metal ’ ; S3 ; ‘ a cannot float on water \ S2 and S3 are
consequences of Si, and, specifically, S2 is an L-consequence, but
S3 is not, and is therefore a P-consequence. Let S4 run : ‘ In this
vessel b of volume 5000 c.c. there are 2 grm. of hydrogen under
such and such a pressure’; S5: ‘In b (of volume 5000 c.c.) there
are 2 grm. of hydrogen at such and such a temperature.* S4 and Ss
are consequences of one another, and, specifically, P-consequences,
since each of these two sentences can be inferred from the other by
means of the physical laws. S4 and S5 are equipollent, but not L-
equipollent, and therefore they are P-equipollent. If in the material
mode of speech we ask whether S3 (like S2) is implicit in Si and
whether S4 and Sg mean the same or not, these questions are
ambiguous. The answer is dependent upon what is legitimately
presupposed in ‘being implicit in*. If we assume only logic and
i86
PART IV. GENERAL SYNTAX
mathematics, then the questions are to be answered in the negative ;
but if we assume the physical laws also, then they must be answered
in the affirmative. For instance, in the latter case 64 and S5 mean
the same to us even if we know nothing more about the described
volume of gas. The material difference between the two assumptions
corresponds to the formal difference between equipollence (in a
P-language) and L-equipollence.
The view that the terms ‘ analytic ’ and ‘ contradictory * are purely
formal and that analytic sentences have the null content has been ex-
pressed by Weyl \K(mtinuurn\ pp. 2, 10 ; he says that a logically con-
tradictory judgment “is recognized as untrue independently of its
material content, and solely on the grounds of its logical structure § ** ;
“judgments which are true purely because of their formal (logical)
structure (and thus possess no material content) we call (logically)
self-evident**. Later, Wittgenstein made the same view the basis of
his whole philosophy. “ It is the characteristic mark of logical sen-
tences that one can perceive from the symbol alone that they are
true ; and this fact contains in itself the whole philosophy of logic **
{[Tractatus^ p. 156). Wittgenstein continues : “And so also it is one of
the most important facts that the truth or falsehood of non-logical
sentences can not be recognized from the sentences alone.** This
statement, expressive of Wittgenstein*s absolutist conception of
language, which leaves out the conventional factor in language-
construction, is not correct. It is certainly possible to recognize
from its form alone that a sentence is analytic; but only if the
syntactical rules of the language are given. If these rules are given,
however, then the truth or falsity of certain synthetic sentences —
namely, the determinate ones — can also be recognized from their
form alone. It is a matter of convention whether we formulate
only L-rules, or include P-rules as well; and the P-rules can be
formulated in just as, strictly formal a way as the L-rules.
{b) VARIABLES
§ 53. Systems of Levels; Predicates
AND Functors
By a system of levels in S, we understand an ordered series
of non-empty classes of expressions which fulfil the six conditions
given on p. 188. Since the number of the expressions of a lan-
guage is, at the most, denumerably infinite, the number of the
classes of is likewise at the most denumerably infinite. These
classes we call levels ; let them be numbered with the finite — and,
if necessary, also with the transfinite — ordinal numbers (of the
§53- SYSTEMS OF LEVELS ; PREDICATES AND FUNCTORS 1 87
second number-class): level 0 (or the zero level), level i, 2, ... w,
a>+ I, — We shall designate the expressions which belong to the
classes of 5Ri by ‘ 0tu * \Stufe^ ; and, specifically, those which belong
to level a (where ‘ a ’ designates an ordinal number) by ‘ “Stu *. [For
the sake of brevity, the phrase “ in relation to is omitted here
and also in the case of the other defined words and the Gothic
designations which follow.] We count all the symbols 0tu as
principal symbols.
An ordered series of w + i expressions 2fi,9l2» ••• 91^+1 (which
may also be empty) is called an expressional framework (^Hg)
\Ausdrucksgerust\ — more precisely, an 7w-termed expressional
framework (9Ig^) — for a particular expressional form if there
exists at least one expression of this form which can occur as a
partial expression in a sentence and is composed of the expresf^ions
2li, ... of the framework, say ^Ig^, together with m principal
expressions 91^,912* ••• alternating order. Thus 9ln has the
form 9li9li9l29l2...9I,„9l49lm+i- The expressions 91'/, . . . 91,'^ are
called the first, . . . mth argument of 9lgi in 91^ ; the series which
they form (in the correct serial order) is called the m-termed
argument-series (9trg or, more precisely, 9(rg^'‘) of 9Igi in 9I„
is also designated by ‘9lgi (9Ii,... 91/,,)'; or, if 9trgi is the series of
those arguments, by ‘ 9Igi (9lrgi) 9In is called a full expression of
9Igi. We say that 9Igi'^ and have the same course of values if
every two full expressions of 9tgi and 9lg2 containing the same
9lrg are synonymous.
The 9Ig^'‘ for the form 0 are called m-termed sentential frame-
works (0g ; 0g^'^) [Satzgeriist]. This is the most important kind
of 9lg. A full expression of 0gi is an 0 ; it is called a full sentence
of 0gi. 0gY is called coextensive with ©g*^ if every two full sen-
tences of 0gi and 0g2 containing the same 9lrg are equipollent.
Theorem 53.1. If 0gi and 0g2 have the same course of values,
then they are coextensive; the converse is not always true (com-
pare, however. Theorem 65.46).
Let 9IgY be composed of “0tUi with or without subsidiary
symbols; let ?(„ be the full expression 9lgi(9lrgi); let here every
argument, as well as 9In itself, be either an 0 or a ^0tu with ^ <a.
Then 9I„ is also called a full expression of 0tUi ; 9Irgi is also called
an argument-series of 0tUi in 91^; StUi is called (in 9l„) m-
termed (0tu^) ; we then designate 91^ also by * 0tUi (9lrgi^ *. If in
i88
PART IV. GENERAL SYNTAX
this case is an Sg, and therefore 9ln an S, then StUi is called
a predicate-expression (^r, ; a symbol is called a
predicate (pr, pr”*, ®pr). On the other hand, if is an Stu, then
StUi is called a functor-expression (gu, gu^, “gu) ; a symbol gu
is called a functor (fu, fu^, “fll). and which are isogenous
and thus of the same level, are called coextensive if the corre-
sponding Sg are coextensive. We say that gUi and gU2, which
are isogenous and thus of the same level, have the same course of
values if the corresponding 9Ig have the same course of values.
The ®Stu are called individual expressions and, as symbols, in-
dividual symbols.
Theorem 53.2. {a) If and ^r2 are synonymous, they are also
coextensive, (d) If gUi and gUg are synonymous, they have the
same course of values. The converse of either is not always true.
(Compare, however. Theorem 66.1.)
and ^r2 are synonymous only if every sentence Si is
equipollent to other hand, they are coextensive if
merely for every full sentence 61 the same condition is fulfilled.
It is possible for ‘P’ and ‘Q* to be coextensive but, for a particular
*pr ‘ M *, the sentences ‘ M (P) * and ‘ M (Q) * not to be equipollent, so
that ‘ P * and * Q * are not synonymous. (In this case, ‘ M (P) ’ is in-
tensional in relation to ‘ P See § 66.)
Conditions: (i) An Stu is not an S. (2) If is isogenous with
an “Stu, then ?Ii also is an “Stu. (3) Every “Stu where a >0 is
either a or an gu. (4) For every ®StUi, there exists a with a
full sentence of which StUi is an argument. (5) Let StUi be an
“Stu where a is greater than i, and which is therefore either a
or an gu. (a) There exists a greatest ordinal number less than a,
say (so that a = + i) ; then for that or gu StUj there exists a
full expression ?Ii such that one of the arguments or 3Ii itself is a
^Stu. {b) There is no greatest ordinal number less than a (for
instance, where ol = w)\ then for every P which is less than a there
is a y such that j8 < y < a, and a full expression for StUi such
that one of the arguments or 9Ii itself is a ^Stu. (6) 9li is as great
in extent as possible, that is to say, the class Stu in relation to Stj
is not a proper sub-class of the class Stu in relation to a series
9?2 which likewise fulfils conditions (i) to (5). — ^9li is called a
suitable argument in general (or for the ith argument-place) for
§ S 3 - SYSTEMS OF LEVELS; PREDICATES AND FUNCTORS 1 89
or (later also for Sflli or if there exists either a
full expression or a full sentence in which occurs at some argu-
ment-place (or at the ith place, respectively).
Examples: In Language II (as in all the usual languages with
higher functional calculus) there is exactly one system of levels. I'o
this the 3 belong as and also the "iPr and gu. The terms ‘ ^Pr’
and ‘ gu* which arc defined here in general syntax are, however,
wider than those previously applied in Language II. According to
the new terms, the ocrfii are ^pr^; ‘ ^ — ’is a ‘ ’ is a Further,
‘ = ’ is a pr^ ; let it be pti ; it is an ‘‘^pr since for every integer w ( > 0)
there exists a full sentence pti('*pr, "pr) (e.g. ‘P = Q’). If we were
to specify that the symbol ‘c’ for the different types (Def. 37.10)
should not be furnished with the corresponding type index, but that
it should be used for all types of irrespectively, then ‘ C * would
also be an ^pr^. Under like conditions ‘ V in ‘FvG’ (Def. 37.5)
would be an
In [Princ. Math.'] Russell has used the symbol ‘ C * and many
others with arguments of any (finite) level whatever, so that,
according to our definition, they belong to the level a>. Russell does
not, however, attribute a transfinite level to these, but* interprets
their mode of use as “systematic ambiguity ”. Hilbert [Unendliche]y
p. 184, and Godel [Unentscheidbare]^ p. 191, were the first to point out
the possibility of introducing transfinite levels.
§ 54. Substitution ; Variables and Constants
What is a variable? It has long been recognized that the old
answers “a varying magnitude’* or “a varying concept” are in-
adequate. A concept, a magnitude, a number, a property — none of
these can vary (although a thing can, of course, have different pro-
perties at different times). A variable is, rather, a symbol with a
certain property. But what property.? The answer: “a symbol
with a varying meaning ” is equally inadequate. For a variation in
the meaning of a symbol is not possible within one language; it
constitutes the transition from one language to another. More
correct is another answer which is frequently given: “A symbol
with a determined meaning is a constant, and one with an unde-
termined meaning is a variable.” But even this is not quite cor-
rect. For it is possible to use constants which have undetermined
meanings; these differ essentially from the variables in that they
do not permit of substitution.
Examples: In a name-language, in addition to names with de-
termined meanings, such as ‘Prague*, names w'ith undetermined
1 90 PART IV. GENERAL SYNTAX
meanings, such as ‘ a ’ and ‘ b *, may also be used. If ‘ Q * is a constant
pr (whether of determined or undetermined meaning makes no dif-
ference), then from ‘Q(jc)’ the sentences ‘Q (Prague)’, ‘Q(a)*,
‘Q(b)* and so on are derivable, but they are not derivable from
‘ Q (a) *. This shows that while * jc * is a variable, ‘ a in spite of having
an undetermined meaning, is a constant. In niaterial interpretation :
‘ a * designates a certain thing ; ix is merely not stated for the moment
(but may, however, be stated later) what thing it designates. In the
examples to be found in this book, constants with undetermined
meanings have frequently beenused; for example,* a ’,‘b’ onpp. I2f.,
*P* and ‘Q’ in many places, such as pp. 25 and 47. The difference
between the variable ' p ’ and the constant of undetermined meaning
‘A* is brought out especially clearly in the examples on p. 158.
Variables and constants are distinguished from one another by their
syntactical character; variables are the symbols of S for which,
according to the rules of transformation of S, under certain con-
ditions, substitution is permissible. This rough distinction is true
for all the ordinary symbolic languages. The exact definition of
‘variable*, however, cannot be so simple, inasmuch as it must take
into account the various possible kinds of substitution, and es-
pecially the three principal kinds — substitution for free variables,
for bound variables, and for constants.
W. V. Quine (in a verbal communication) has shown that it is
possible to use an operator-constant instead of an operator-variable.
Instead of ‘ (jc) (jc — jc) ’ we can, for example, write ‘(0)(0i=0)*.
Incidentally, we can extend this method so that a language (even a
language which includes both arithmetic and infinitesimal calculus)
contains no variables at all. For instance, in Language II we may,
to begin with, construct a Language IT in which no free variables
appear in sentences. Here PSII i6 and 17 have to be replaced by
rules of substitution: (t)i) (Si) may be transformed into »
arid (Pi) (^i) irito 61 RII 2 drops out; but several new
rules must be formulated. IT is then constructed from IT by writing
instead of a bound Ui, in the operator and in the places of substitution
in the operand, some expression or other from the range of values of
Di- [In IT, as opposed to the usual languages, related symbols are
always isogenous.] In the symbolic languages hitherto in use,
substitution for constants does not occur. Languages of the kind
indicated, with no variables (but having constants as variable-ex-
pressions) must be differentiated from languages without substitution
(that is to say, with no variable-expressions whatever). See the
example of Ijc on p. 194.
Examples of the three principal kinds of substitution: in I and II
§ 54- SUBSTITUTION ; VARIABLES AND CONSTANTS I 91
‘2 = 2* is derivable from ‘jc = jc’; in II from ‘(jc)(jc = jcV; and in IV
from‘(3)(3 = 3)*.
We say that substitution occurs in S when there are expres-
sions in S — ^we call them variable-expressions (93) — to which what
now follows is applicable, and which, in particular, fulfil the con-
dition given below, p. 195. [This condition can at that point be
formulated more simply with the help of the terms which will by
then have been defined.] [To facilitate the comprehension of
what we are about to say, it should be noted that in the ordinary
symbolic languages, all 93 are symbols, and, specifically, variables. ]
To every 93, say 93i, there is correlated a class (which may also be
empty) of expressions which we call operators (Op), or, more pre-
cisely, operators with 93i (OpjBi)- Let Opi be an Opsp^ ; then there
is correlated to OPi a class of principal expressions which we call
substitution- values of 93i in relation to Opi ; this class contains at
least one expression which is not synonymous with 93i. Further,
to 93i itself is correlated a class of principal expressions ^hich we
call substitution-values for free 93i; this class, when it is not empty,
contains at least two expressions which are not synonymous with
one another. Let be that class to which belong all substitution-
values for free 93x and all substitution-values for in relation to
some CpsBj, together with all expressions which are isogenous with
one of the above. We call the expressions of the values of 93i.
OPi®! is called unlimited if every value of 93i is also a substitution-
value of 93i in relation to Opj ; otherwise, limited.
Let 9Ii be a full expression of jS and specifically either an 0
or an Stu ; and let 9Ia be constructed from 9Ii by replacing every
argument 91^ (« = i to m) by a 93^ to the values of which 91^ belongs,
91^ being so qualified that it can occur as a partial expression in
a sentence. 9I2 is then called an m-termed expressional function
(9Ifu, 9lfu"*) ; 93i is called the ith argument in 9I2. An 9lfu"* is
called improper when m = 0; proper when m>0. If 9Ii is here an
0, then 9I2 is called an m-termed sentential function (0fu, 0fu"*).
The 0fu constitute the most important kind of 9Ifu.
The difference between sentential framework, sentential function, and
predicate-expression should be carefully noted, since, owing to the
fact that the term ‘ sentential function * is used in all three meanings,
this difference is often disregarded. Examples of Sg in II (here we
separate the expressions of the expressional series by dashes) :
PART IV. GENERAL SYNTAX
192
‘P(3, — )vQ( — )*, ‘Q( — )* [but also ‘(A) — (B)* (with the ar^-
ment ‘ v *) and ‘ ( — x)(P(jc)) * (to which ‘ 3 ’ and the null exprdssion
are the suitable arguments)]; examples of Sfu: ‘P(3, Jc) vQ(x)*,
‘Q(^)*; examples of ^r: ‘Q* [but also ‘sm(P,Q)* — see p. 86],
The differences between the remaining the remaining ^fu, and
the ($u are analogous. The only reason why we must also deal with
the ?lg and the 0g in addition to the ^fu and the SfU is that it
cannot be generally assumed that there are in every language variable-
expressions for the arguments concerned.
The Sfu° are 0. In I and II, all the 0fu are 0, and, specifically,
the proper 0fu are open 0, and the improper 0fu are closed 0.
In the majority of the usual symbolic languages, all the 0fu are 0 ;
in many of them, however, the rules which govern this point are
not clear.
Let Opi occur at a certain place in Sj; then, to this Opi is
correlated by means of definite rules of formation (which, like
all rules of formation, are contained in the rules of trans-
formation; see above), a partial expression ?lfu of Si consist-
ing of Opi, an SfUi, and sometimes subsidiary symbols as
well; SfUi is called the operand of Opi (at this place) in Si.
[Usually, SfUi here comes after Dpi ; and sometimes the begin-
ning, the end, or both, of the operand SfUi is indicated by means
of special subsidiary symbols (for example, by brackets in I and II
and by dot-symbols in Russell) as well as by OPi-] We designate
9lfUi also by ‘Opi (SfUi)’. If SfUi can be an operand belonging
to Dpi — that is to say, if there exists an 2lfu of the form Cpi (SfUi)
— we call SfUi operable in relation to Dpi- 35i is called bound in
^{2 at a particular place if this particular place belongs to a partial
expression of which has the form Dpi©^ (Sfu); and, speci-
fically, it is called limitedly (or unlimitedly) bound if Opi is
limited (or unlimited, respectively). If occurs in 9I2 at a
place at which 93i is not bound, then 93i is called free at this place
in 9I2. The places at which occurs freely in 5I2 are called sub-
stitution-places for in designate by ‘ 2tfUi
the ex-
pression which results from 3lfUi on replacing 33i by at all sub-
stitution-places in 9IfUi ; here must be a value of 93i, and there
must be no 982 which occurs freely in 9li and is bound in 9lfUi at
one of the substitution-places for 93i. [If does not fulfil these
conditions, or if 95i does not occur freely in 9IfUi, then ‘ 9lfUi
»93
§ 54. SUBSTITUTION ; VARIABLES AND CONSTANTS
designates itself.] We call a variant of 9IfUi
(in Si). A sentence of the form is called a variant of
SfUl in relation to Opj®^ if 0fUi is operable in relation to and
9li a substitution-value of ®i in relation to Opj.
We distinguish between two different kinds of operators:
sentential operators and descriptional operators. If Op^ (SfUi) is
an Sfu, say SfU2, then Opi is called a sentential operator in SfU2 ;
and if every expression of the form Op^ (Sfu) is an Sfu, then
Cpi is called a sentential operator. Assume that Dpi (SfUi ) is not
an Sfu, and is hence another ^fu, say ^fU2 *, then 9 lfU 2 is called a
descriptional function^ or, if it is closed, a description. A description
is, accordingly, always an Stu. Opi is then called a descriptional
operator in ^fu 2 ; and if every expression of the form Opi (Sf U )
is a descriptional function, Op^ is called a descriptional operator.
Let Si be Dpi®^ (SfUi); Opi is accordingly a sentential operator
in Si. If, here, ®i occurs freely in SfUi, and if every variant of
SfUi in relation to Opi is a consequence of Si, then Opi is called
a universal operator in Si. If Opi®i is a universal operator in every
sentence of the form Opi (SfU 2 )» where SfU 2 is any Sfu in which
®i occurs freely, then Opi is called a universal operator.
Let ®i occur freely in Si ; then, if every variant Si
where
?l 2 is any substitution-value whatsoever of free ®i, is a conse-
quence of Si, we say that in Si there exists substitution for free
®i. If in every sentence in which ®i occurs freely, substitution for
free ®i exists, then we say that there exists (in S) substitution for
free ®i.
For the foregoing definitions, beginning with *, it is required
that the following condition be fulfilled: namely, for every ®i
there is at least one Si such that either there exists substitution for
free ®i in Si, or Si has the form Opi®^ (3fUi), where ®i occurs
freely in SfUi and OPi is a universal operator in Si.
9li is called a substitution-value of ®i if at least one of the
following conditions is fulfilled: (i) There is in S substitution for
free ®i, and 2li is a substitution-value for free ®i ; (2) There exists
in S a universal operator OPis^ and 2li is a substitution-value in
relation to Opi.
SL
14
194
PART IV. GENERAL SYNTAX
If 95 i occurs freely in Si, but if at the same time there exists no
substitution for free ®i in Si, then we say that ®i is constant in Si
(in the usual languages this does not occur). If 33 i is constant in
every sentence in which it occurs freely, and if at least one such
sentence exists, then we call 93 i constant. If ai is a 93 and constant
(either in Si or generally), then we call Qi (either in Si or gener-
ally) a variable-constant ; if Ui is a 93 and constant in no sentence,
then Qi is called a variable (d). All symbols which are 93 , and hence
all 0 also, are counted amongst the principal symbols. If Qi is not
a 0 (and hence either not a 93 at all or a 93 which is constant in at
least one sentence), Ui is called a constant ({). If fi is an "Stu,
then fi is called a constant of the level a (“f).
Si is called open if there exists a 93i such that it occurs freely in
Si and there is substitution for free 93 i in Si; otherwise. Si is
called closed. An 9 Ii which is not an S is called open if there
exists a 93 i and an Si such that 9 li is a partial expression of Si,
93 i occurs at a place in 9 Ii at which it is free in Si, and there
is in Si substitution for free 93 i; otherwise, 9 Ii is called closed.
If no substitution for free 93 exists in S, then all 91 are closed;
S is then called a closed language-system.
Example of a closed lang^ge-system: IT, p. 190.
A language-system without variable-expressions can easily be
constructed; obviously such a system is also a closed system. An
example is afforded by Language Ik, which is constructed in the
following way as a proper conservative sub-language of I. Symbols
of Ik are the f of I. The 3 (and 6) of Ik are the 3 (and S, respec-
tively) without 0 of I. As schemata of primitive sentences, PSI 1-3
remain unchanged, PSI 4-6 and ii drop out, PSI 7-10 are replaced
by the following: 7. 3 i= 3 i. »• ( 3 i= 32 ) 3 (®. 3 ex[||]).
9. '^(nu= 3 i')- lO- ( 3 i' = 32')^ ( 3 i= 32)- Of the rules, RI 2 and
3 remain unchanged ; RI i and 4 drop out. The definitions are not
formulated as sentences, but as syntactical rules concerned with
synonymity. All the definitions in I can be correspondingly trans-
ferred to Ik. For instance, in place of D 3 (p. 59) the rule is
given: “ If fUi is ‘prod*, then for any 32 » fUi(nu, 3 a) is synonymous
with nu, and for any 3i and 3ai f'*i(3i'»3a) is synonymous
with fu, [fUi( 3 i, 32)1 32]. where is ‘sum*.** To a syntactical
sentence concerning an open sentence of I, there corresponds a
syntactical sentence concerning sentences of Ik of a particular form.
For instance, the sentence : “ Every sentence of the form fUi( 3 i, 32)
= fWi(32»3i)» where fUi is ‘prod*, is demonstrable in Ik** corre-
§ 54 - substitution; variables and constants 195
sponds to the sentence: ‘“prod(A;,y) = prod(y,jc)’ is demonstrable
in I.” In this way, arithmetic can be formulated in It must
nevertheless be noted that here 33 are only given up in I^ itself ; for
the syntax-language, on the other hand, 33 are necessary in order to
formulate the primitive sentences and rules as general stipulations.
If Si is closed and contains no Sfu (and hence no S) as proper
parts, then we call Si an elementary sentence. In an elementary
sentence, neither 0, Dp, nor 35f (§ 57) occur.
If D is an fa, then 0i is called a sentential variable (f). If all
substitution-values of 0i (in Si or in general) are "ipr, then Di is
called a predicate-variable (p) (in Si or in general, respectively) ;
if all substitution-values are then Ui is called a p"^. The same
applies to the jju : functor -variable (f, f ”‘). Let all the substitution-
values of 33i (in Si or in general) be “Stu. Then S3i is called (in Si
or in general) an “3S (correspondingly “d, '^p, “f). A is called an
individual variable^ a an individual constant. Let all the substitu-
tion-values of 33i (in Si or in general) be Stu, but of various levels ;
then 33i is called (in Si or in general, respectively) an ^^^33 if for
every jS < a there exists a y such that ^ y < a, so that at least one of
the substitution-values (in Si or in general) belongs to the level y,
but none to the level a or to a higher one. [According to this, for
example, in the Sfu, ' 5 Pri(Pi), Pi is an if and only if 'pii is an
“33i is not necessarily an Stu; “^SSi is an Stu (in Si or in
general) and, more precisely, an “Stu if and only if 93i occurs
freely (in Si or in at least one S respectively). An ^“^33 is not an
Stu.
Examples: i. Languages I and II. All 33 are D. ‘‘d are the 3.
Substitution-values for free *x' are the 3 1 substitution-values for
in relation to ‘(3:«)2(P(x))* are the 3 which are synonymous
with ‘O’, ‘ I ’, or *2*. Every p (or f) is an Stu of a certain level;
values and substitution-values are all 3?!^ (or 5u, respectively) of the
same type. Sfu are the S. Every S is operable in relation to every
operator. Substitution for free u: ‘P(3)’ is a consequence of
‘P(a:)’; for bound o: ‘P(3)’ is a consequence of ‘(jf) 5 (P W)’-
Sentential operators are the universal and existential operators;
descriptional operators are the K-operators. — 2. In Russell s lan-
guage, there are descriptions which are ®Stu, and also des,criptions
which are SPr. For instance, ‘^(P(x))* is a class-expression and
thus a ; it is a description with the descriptional operator x .
Correspondingly, ' xy' is a descriptional operator for a
196
PART IV. GENERAL SYNTAX
§ 55. Universal and Existential Operators
We shall first discuss the subject in the material mode of speech.
Let a domain contain m objects, and a certain property be attributed
to each one of these objects by means of the sentences Si, S2, • • • Sm
respectively. Now if S„ mean^ at least as much as the sentences
Si to Sto taken together, we may call Sn a corresponding universal
sentence in the wider sense ; and specifically, if Sn does not mean
more than all the individual sentences put together — that is to say,
if it means exactly what they do — a proper universal sentence. If
the universal sentence is constructed with a universal operator,
then the closed variants of the operand are the corresponding in-
dividual sentences. We therefore define as follows: a universal
operator Dpi (restricted or unrestricted) is called a proper unu
versal operator if every closed sentence of the form Dpi(SfUi), for
any 6fUi whatsoever, is a consequence of (and hence equipollent
to) the class of ' ic closed variants of SfUi in relation to Opi;
otherwise it is called an improper universal operator (namely, if
there exists a closed sentence OPi(6fU2) which is not a conse-
quence of the class of the closed variants of Sfu2 in relation to
OPi).
An existential sentence follows from every one of the individual
corresponding sentences. Materially expressed, its meaning is con-
tained in the meaning of each of the individual corresponding sen-
tences and therefore also in the conunon meaning. If the sentence
means no less than this common meaning, but precisely the same,
then it may be called a proper existential sentence. Hence we
define in the following manner.
Let be Opisoi (®fUi) ; Opi is accordingly a sentential operator
in ; if here 93^ occurs freely in ®fUi and if is a consequence
of every variant of Gfui in relation to Op^, then Dpi is called an
existential operator in If Dpis^ is an existential operator in
every sentence of the form Opi(SfU2), where Sfu2 is any Sfu in
which 93 i occurs freely, then Dpi is called an existential operator.
[This definition is analogous to that of * universal operator’ on
p. 193.] Let DPi be an existential operator. If the content of each
closed sentence of the form OPi(SfUi) coincides with the pro-
duct of the contents of the closed variants of ®fUi in relation to
Opi, then Opi is called a proper existential operator; otherwise it
§ 55- UNIVERSAL AND EXISTENTIAL OPERATORS 1 97
is called an improper one (namely, if there exists a closed sentence
OPi(SfU2) whose content is a proper sub-class of the product
of the contents of the closed variants of SfU2 in relation to Opi).
Examples: Universal operators occur in the languages of Frege,
Russell, Hilbert, Behmann, G 5 del, and Tarski (see § 33); they have
in the majority of cases the form (0). Existential operators also occur
in each of these languages; in those of Russell, Hilbert, and Beh-
mann, some are simple (for example, formed either with ‘ 3 ’ or with
‘E*), but in all such languages there are operators composed of two
negation-symbols and a universal operator which are not usually
called existential operators. (In II, for instance, ‘ is
also an existential operator.) In the languages mentioned, the simple
universal and existential operators are unlimited; but it is also
possible to construct limited operators [such as ‘(je)((jc<3)D ’
and ‘( 3 ^)((^< 3 )**]* Languages I and II there are also limited
operators which are simple, that is to say, which contain no partial
sentence.
In Languages I and II the universal operators with 5 are proper
universal operators. For not only is every sentence — and hence
every closed sentence — of the form ptiO) a consequence of
(3i)(pri(3i))» l^ut, conversely, this universal sentence is also a conse-
quence of the class of those closed sentences (by DC 2, p. 38) and
therefore equipollent to it. In the other languages which we have
mentioned, on the contrary, the same thing is not true for the uni-
versal operators with or with 3 (unless Hilbert's new rule is laid
down; see p. 173); hence these operators are improper.
The universal and existential operators of higher levels — that is to
say, with p (or f) — are apparently improper in the majority of
languages. In the case of the earlier languages, this follows from the
same cause as before, namely, from the lack of indefinite rules of
transformation. But in the case of Language II, it is true for a dif-
ferent reason. For the sake of simplicity, we will restrict ourselves
to the logical sub-language II{ of II. Let the ^pihprj, of IIi desig-
nate (in material speech) a property which belongs to all the
number properties which are definable in IIi but, on the other hand,
not to all the number properties which are indefinable in II i (see
p. 106). Then (Pi)(pr2(Pi)) is contradictory; the class of all closed
variants of the operand is, however, analytic ; and hence this contra-
dictory sentence cannot be a consequence of it. Further, on the
same hypothesis (3Pi)('^pr2 (Pi)) is analytic ; here all closed variants
of the operand are contradictory ; the content of the existential sen-
tence is null and the product of the contents of the variants is the
total content ; therefore the former is a proper sub-class of the latter.
Let occur at a certain place in and let it be either free or
bound by Op^. Let in the first case be the class of substitution -
values of a free 95i ; and in the second case the class of substitution-
198 PART IV. GENERAL SYNTAX
values of 33 i in relation to Opj. Let be subdivided into the
largest sub-classes (non-empty) of expressions synonymous with
each other. We call the number of these sub-classes the variability-
number of 93i at the place in question in Qi ; in the case of a finite
(or infinite) number we speak of finite or infinite variability re-
spectively. We say that 33 ^ at a certain place in Sj has infinite
universality if 93i has infinite variability at that place, and is there
either free or bound by a universal operator.
Examples: has in ‘( 3 ^) 5 (P(^))* the variability-number 6;
in ‘P(3c)* and in ‘(x;)(P(x))’ it has both infinite variability and
infinite universality. In a sentential calculus of the usual form,
with only free j, and no constants la, every sentence is either analytic
or contradictory. Thus every f there has the variability number 2.
The same is true even when we introduce universal and existential
operators ; the [ are then unrestrictedly bound but have only finite
variability.
We call, 5 li a greatest definite expressional class if the following
conditions are fulfilled: (i) For every of there is a sentence
which is capable of being sub-divided into expressions of of
which is one; (2) If Si is determinate and capable of being sub-
divided into expressions of 5^i, and if Si contains no expression
with infinite variability, then Si is resoluble; (3) fti is not a proper
sub-class of an expressional class which likewise fulfils conditions
(i) and (2). We call the product ftg greatest definite
classes of expressions of S the definite expression-class of S. Si is
called definite if it is capable of being sub-divided into expressions
of 5^2 2nd contains no expression having infinite variability ; other-
wise, indefinite, [The terms ‘ definite ’ and ‘ indefinite ' hereby de-
fined are themselves indefinite. Before, in the syntax of I and II,
we defined the terms ‘ definite ^ and ‘ indefinite ’ as definite terms ;
such definitions cannot be formulated generally, but only specifi-
cally, for particular languages — that is, if they are to express
approximately the meaning which is intended (cf. § 43). The terms
‘ definite ’ and ‘ indefinite ’ as defined here will not be used in what
follows. If in general syntax the word ‘definite* or ‘indefinite*
occurs in relation to the syntax-language (as, for instance, on
p. 171), we may look upon Language II (or some kindred lan-
guage) as the syntax-language and take the earlier definition of
‘definite’ (§ 15).]
§ 56. RANGE 19^
§56. Range
(Compare the addition at the end of § 57)
We have called complete if everj’ sentence is dependent upon
Sii- A complete R leaves, as it were, no question open ; every sen-
tence is either affirmed or denied (though not, generally, by a
definite method). If is contravalid, then is complete in a
trivial sense : every sentence is at the same time affirmed and denied.
We will call Ri2l premiss-class if is complete but not contravalid,
and if there exists no complete class which is a proper sub-class
of Ri.
Theorem 56.1* (a) If S is inconsistent (§ 59), then there are no
premiss-classes in S. (b) If S is consistent and logical, then the
empty sentential class is the only premiss-class, (c) If S is* de-
scriptive (and therefore consistent), then every premiss-class is
both non-empty and indeterminate, and every one of its sentences
is indeterminate.
Theorem 56.2. Two non -equipollent premiss-classes are always
incompatible with one another.
In material interpretation, every non-empty premiss-class re-
presents one of the possible states of the object-domain with which
S is concerned. is called a premiss-class of 5^2 — in the sense of
a correlate of ‘ consequence-class * — if is a premiss-class and SI2
a consequence-class of 5 Ii. That fti is a premiss-class of Sj means,
in material interpretation, that is one of the possible cases in
which Si is true. By a range we understand a class of premiss-
classes such that each class which is equipollent to a premiss-class
belonging to Sflj, belongs also to JUj. By the range of R^ we under-
stand the class of premiss-classes of R^. That is the range of 0 i
means, in material interpretation, that 9Ki is the class of all possible
cases in which Sj is true ; in other words, it is the domain of possi-
bilities left open by
Herein lies the reason for the choice of the term ‘ range * (‘ Spiel-
raum')\ in adopting it we, have followed Wittgenstein [Tractatus^y
4.463, p. 98: “The truth-conditions determine the range which is
left to the facts by the proposition.” Wittgenstein, however, does
not give a syntactical definition.
By the total range we understand the class of all premiss-classes.
The terms ‘range’ and ‘content’ to some extent exhibit a
200
PART IV. GENERAL SYNTAX
duality, as is shown, for example, by the following theorems
(3 to 6) which are analogous to theo’^ems 49.1, 2, 4 and 5.*
Theorem 56.3. If a consequence-class of 5 ti, the range of
is contained in that of 512-
Theorem 56.4. If Ri and 5I2 are consequence-classes of one
another, they have the same range.
Theorem 56.5. If 5 li is valid, the range of 5 li is the total range.
Theorem 56.6. If Ri is contravalid, the range of Ri is null.
Theorems 3 to 6 hold correspondingly for Si and 62-
Theorem 56,7. (a) The range of 5 li + 5^2 is the product of the
ranges of 5 Ii and 5^2- (b) The range of 5 ti is the product of the ranges
of the individual sentences of R^,
By the supplementary range of 5 ti, we understand the class of
premiss-classes which are not premiss-classes of Ri. The supple-
mentary range of R^ is always also a range ; but it is not always the
range of a 51 . If the supplementary range of 5 li is the range of 5^2*
then we call 5^2 a contra-class to 5 li. Correspondingly, Sg i® called
a contra-sentence to Si if {S2} i^ ^ contra-class to (Si). If S2 is
a cemtra-sentence to Si, then Si is likewise a contra-sentence to
S2. If S2 is a contra-sentence to Si, then, in material interpreta-
tion, S2 is true in all the possible cases in which Si is false — and
only in these ; thus, S2 means the opposite of Si. If, in S, there is
no negation, then, as a substitute for Si, we can take a contra -
sentence to Si, or a contra-class to { 0 i}. In case neither exists,
then there is no substitute for ^ Si, but there is a substitute for
the range of Si, namely, the supplementary range of Si, there
being always exactly one such range. — The terms ‘ range * and ‘ sup-
plementary range ’ will make, it possible for us to characterize the
individual sentential junctions.
§ 57. Sentential Junctions
If there is a full sentence Si of Sg”, in which all n arguments
are S, then Sg? is called an n-termed sentential junction in Si.
• (Note, 1935.) It is, however, to be noted that the converses of
Theorems 3-6 do not generally hold ; this fact has been pointed out
to me by Dr. Tarski. To ascertain the exact situation a further de-
tailed investigation is required. In particular, it would be worth
while to search for a different deftnition of ‘ range * which secures
duality in a higher degree.
§57* SENTENTIAL JUNCTIONS 201
If Sg?, with n arbitrary sentences as arguments, constitutes a full
sentence, then is called an n-termed sentential iuncdon
( 95 f, Sf"). If SQi is composed of and possibly subsidiary
symbols as well, is called an n-termed sentential predicate-
expression; if Q] is a sentential predicate-expression, is called
a sentential predicate, or a junction-symbol (of, ot"). A nl” is,
accordingly, a ^pr" to which sentences are suitable as arguments.
In order to prepare for the definitions of particular kinds of junc-
tion, we will proceed in a way that is dependent upon the method
of the value-tables (see § 5), but without assuming that S contains
a negation. Let us consider a value-table for, say, three members,
®i» S2, and S3. The second row runs: ‘TTF*; and to the case
designated by this row corresponds the sentence Si •02* /•w S.-
Let S4 be any junction sentence Sfi (Si, S2, S3). For this the
column in the value-table may be stated ; in the second row it is
occupied either by ‘T* or by ‘F*. ‘T* would mean that S4 was
true in the second case, and that, accordingly, S4 was a conse-
quence of Si« S2« ^ S3; ‘ F* would mean that S4 was false in the
second case, and that therefore ^S4 was a consequence of
Si#S2«^S3. We want now to express these relations without
making use of negation, and this is possible with the help of the
ranges. We will (in this section only) designate the range of Si by
* [SJ * and the supplementary range of Si by ‘ — [SJ *. Si • S2
has the same content, and thus the same range, as {Si, S2}. Hence,
according to Theorem 56.7 A, [Si«S2] is the product of [SJ
and [SJ. We replace the range of ^Ss by — [®3]; hence we
replace [Si«62»'^®3] product of the classes [SJ,
[S2], — [S3]. That S4 (or ^S4) is a consequence of this con-
junction is (according to Theorem 56.3) expressed by the fact that
[Si#S2«'^S3] is contained in [SJ (or in — [SJ, respectively).
On the basis of the foregoing conclusions we can now state the
following definitions.
Let Si, S2, • • • S,i be n closed sentences. We construct (according
to the rows of the value-tables) the m( = 2”) possible series
... of « ranges each, where the fth (1 = i to n) range is
either [SJ or — [SJ. The suffixes of the SR may be determined
according to a sort of lexicographical arrangement of the ranges:
if SRfc and SRj agree in the first i~i serial terms (ranges), while the
ith term of SR^ is [S J and of SRj is — [S J, then SR* must precede
202
PART IV. GENERAL SYNTAX
3li, that is to say, k must be less than /. We will now construct a
series of m ranges, 9Jli to (which likewise correspond to
the rows of the table, namely, to the conjunctions), in such a
way that, for every k {k=^i to m), SJlfc is the product of the
ranges of the series 9?^. If, for a certain 93!i and a certain k
and n arbitrary closed sentences Sj, . . . Sn, the class 991^,
constructed for Sj, ... Sn in the way already described, is always
a sub-class of [23ti(0i, ... 0„)], we say that the /rth characteristic
letter for is ‘ T *. If, on the other hand, for any closed 0^, . . . 0„,
is always a sub-class of — [®fi(0i, ... ®n)]> we say that the
Ath characteristic letter for is ‘ F *. If neither of the two con-
ditions is fulfilled, then does not possess any /:th character-
istic letter. If possesses a characteristic letter for every
^ (/e = I to ;w), wc call the series of these m letters the characteristic
of 93fi. — Let 01, . . . 0,1 be n closed sentences of any kind ; 9Mi,. .
the ranges which are constructed from these in the manner stated.
Then every premiss-class of S belongs to exactly one of these
classes 9R. For any St'i which possesses a characteristic,
[33ti(0i, ... 0„)] is the sum of those 911^ for which the ^th cha-
racteristic letter is ‘T\ — For the 93!'* there are 2^" possible cha-
racteristics.
With the help of the characteristic we are now in a position to
define the various special kinds of junctions ; we will restrict our-
selves here to the most important of these. We call a 93!^ with the
characteristic ‘ FT ’ a proper negation, and a 93f^ with the charac-
teristic ‘TTTF’ (or ‘TFFF\ ‘TFTT’, ‘TFFT*, ‘FTTF’) a
proper disjunction (or conjunction, implication, equivalence,
exclusive disjunction, respectively).
If for every 0i, (0|) is incompatible with 0i, then 93fi is
called a negation, is called a disjunction if for any Si and 02
whatsoever, 9310(01,02) always a consequence of 0i and a
consequence of 00. is called a conjunction if, for any 0i and
02, 01 and 02 are always consequences of Sf3(0i, 00)- is
called an implication if, for any 0i and Sg, 02 is always a conse-
quence of {0,, 1^4(01, 02)}. If a junction of these kinds is not a
proper one, we call it improper. If for one of the junctions men-
tioned there exists a junction-symbol, we call it a symbol of nega-
tion (proper or improper) or a symbol of disjunction, etc., re-
spectively.
§ 57 - SENTENTIAL JUNCTIONS 203
Theorem 57.1. If SSfj is a negation, then for any every sen-
tence is a consequence of {Si, ®fi (Si)}. — The class here mentioned
is contra valid.
Theorem 57.2. If 93 !i is a proper negation, SSIg a proper dis-
junction, and Sfs a proper conjunction, then for any Sj the fol-
lowing is true: {a) If Gj is closed, ®fi(Si) is a contra-sentence to
Si; (*)*; (c) is valid; (d) a)fa( 93 fi(Si), S^) is
contravalid. According to (^:) and (^/), the principles of traditional
logic such as those of excluded middle and of contradiction are valid
in every language S for the proper junctions, if such occur in S.
Theorem 57.3. If Sfj has a characteristic, and if ©fg is co-
extensive with SIi, then 93I2 has the same characteristic.
Examples: The junctions which are designated as ‘negation* are,
in the majority of systems (for instance, in those of Frege, Russell,
and Hilbert, and in our own Languages I and II), negations in the
sense here defined. In I, ‘ '^Prim(;c)* is not a contra-scntencc to
‘ Prim (x) * ; both senten<;cs are contradictory, and their range is thus
null. In spite of this, ‘ ^Prim(:ic)* is equipollent and equal in
range to ‘Prim(jc)*. If ‘Q* is an undefined prt), then in II there
exists a contra-sentence to *Qix)\ namely ‘ '^(^)(Q (.v)) *. In I, on
the other hand, there is neither a contra-sentence nor a contra-class
to ‘ Q (jc) * ; but there is a supplementary range.
In the systems of Russell and of Hilbert and in our own i languages
I and II, ‘ V* is a symbol of proper disjunction. In Hilbcrt*s system,
the junction which consists of three null expressions is also a proper
disjunction (6162 is equipollent to OjV Sg)- In the English lan-
guage the connectives ‘either... or* ('2I3 is empty) — as also ‘aut...
aut* in the Latin language — constitute a proper exclusive disjunc-
tion. Hilbert’s symbol and the of I and II and Russell’s
system (and in the latter also the many-point-symbols) are symbols
of proper conjunction. In Russell and in I and II, ‘D’ is a symbol
of proper implication^ as is also Hilbert’s ‘
In the systems of Russell and Hilbert and Languages I and II, all
bl have a characteristic; but in those of Heyting and Lewis uf with-
out a characteristic also occur. For instance, Heyting’s symbol of
negation (we will here write it thus ‘ — ’) is a sytnbol of improper
negation^ without a characteristic. Si and — Si are certainly always
incompatible with one another ; but — Si is not always a contra-
sentence to Si. Si and — Si possess the common consequence
Si V — Gj, which is in most cases not valid but indeterminate.
Si is not generally equipollent to Si. In Lewis’s system, the
symbol of strict implication is a symbol of improper implication^
* Omitted in this edition.
PART IV, GENERAL SYNTAX
204
without a characteristic (see §69). (Concerning the intensionality
of the d! which have no characteristic, see § 65.)
Let Opi®j be a universal operator^ and an existential
operator \ let the substitution- values of be the same in relation
to OP2 as they are in relation to Opi ; and left be a negation. We
call Dpi, Opj, and Sfj associated^ if for every Sfu^ which is operable
in relation to Opi and Opj, 3 }fi (Opi(SfUi)) is equipollent to
Op2 (®fi(®fUi)). If both the operators as well as the negation are
proper, then they are also associated.
Example: In II, (p^) and (3 Pi) are certainly improper; but these
operators and are associated, since ^(Pi)(6fUi) is always
equipollent to (3 Pi) 6fUi).
{Addition, 1935.) Since the concept of ‘ range * as defined above
does not always fulfil the requirement of duality (see footnote,
p. 200), the definitions of the sentential junctions based upon
this concept are not always in accordance with the usual meanings
of the junctions as laid down by the truth- value-tables. Dr. Tarski
has found simpler definitions of the sentential junctions which do
not make use of the term 'range'. It is possible to proceed, for
instance, as follows. We say that the relationship of negation sub-
sists between Ri and 5^2 if iti and 5^2 are incompatible and have
exclusive contents. 5I3 stands in the relationship of disjunction to
and 5^2 if the content of is the product of the contents of R^
and R^ (compare Theorem 34^.8). ftg stands in the relationship
of conjunction to R^ 'and 5^2 if its is equipollent to Ri + 5^2- ^3
stands in the relationship of implication to fti and R2 if the fol-
lowing two conditions are fulfilled: i. 5^2 is a consequence-class
of Ri+Rq) 2. if R^ is smaller in content than 5^3, then R2 is not
a consequence-class R^’j-R^ (compare §65, paragraph i). R^
stands in the relationship of equivalence to R^ and 5^2 if -$^3 stands
in the relationship of implication both to ftj and R2, and to R^
and — ^We then call ®!} a proper negation if, for every closed
i (®i) stands in the relationship of negation to Si. We call
Sfg a proper disjunction if, for closed Si and S2, 93f2(®i»®2)
always stands in the relationship of disjunction to Si and S2. The
remaining junctions are to be similarly defined.
§58. ARITHMETIC
20s
(c) ARITHMETIC; NON-CONTRADICTORINESS;
THE ANTINOMIES
§58. Arithmetic
Let % be an “Stu, 8 fUi an and 9li the infinite series of
expressions constructed in the following manner : the first term is
3 lo, and for every n the (n + i)th term is the full expression of gui
with the nth term as argument. has accordingly the form 31^;
0rUi(3Io); gUi(?ln); .... If every two dif-
ferent expressions of 3ii are isogenous (hence each one an “Stu)
but not synonymous, we call a numerical expresnon-series or
3-series. The expressions of and those synonymous with them
are called numerical expressions (3) of Those 3 which are
synonymous with are called null-expressions, or 0 - 3 » of Sflj ;
those which are synonymous with 3rUi(3Io) are called 1-3 of Kj,
etc. A 3 which is synonymous with 8fu(3i) is called a^successor^
expression of 3i. [These and the following terms are always re-
lated to a particular 3 -series for the sake of brevity, the phrase
**of or “in relation to SRj” will usually be omitted.]
If Qi is a 3i it is called a numeral ( 55 ). If 0i is a 0-3> it is called
a zero-symbol (nu). is called a numerical 95 if the 3 belong to the
substitution-values of 95i. If Pi is a numerical 93, then Pi is called
a numerical variable ( 5 ).
If, for Sqi (or 1 ), there exists a full sentence with only 3 as
arguments, then Sgi (or ^ti) is called a numerical Sg (or ^r).
If for $Ui there exists a full expression such that this expression
itself and all the arguments are 3 > then gfUi is called a numerical
gfu. If pti (or fui) is a numerical or 3fu, then pti (or fUi) is
called a numerical predicate ( 5 pr) (or a numerical functor ( 3 fu),
respectively).
® 9 i (or ^ti) is called a sumSq (or -^r, respectively) for the
^h place (ife= I, 2 , or 3 ) if for any m and n whatsoever, the fol-
lowing is true : if 3i is an iw-3> and 38 an 71 - 3 , then the full sen-
tence of Sgi (or of ^ri, respectively) in which 38 is the ^th argu-
ment and 3i and 32 other arguments, is valid when and
only when 33 is an (m + 7i)-3. (Juf is called a provided
that gUi is a numerical gu and the following is true for any m and
« : if 3i is an m-3 and 32 ^ (3i> 32) is an {m + 7i)-3.
2o6
PART IV. GENERAL SYNTAX
" Product-’(5^\ ‘-3rU* are analogously defined, where 33*
or 2rUi(3i»32) respectively, is an If 5pri is a sum-^r
(or product-^r), spti is called a sum-predicate (or product-predicate ^
respectively). If 3fUi is a sum-0fu (or product- Ju), 3fUi is called a
sum-functor (or product-functor, respectively). It will readily be
seen that in a similar way all the other arithmetical terms which
occur in the arithmetic contained in S can be syntactically charac-
terized; that is to say, those kinds of Sg, ^r, or gu to which a
particular arithmetical meaning belongs can be defined. We shall
content ourselves here with the foregoing examples.
We say that S contains an arithmetic if, in S, there is at least
one 3-series SRi, one sum-0g, and one product-Sg, in relation to
Let S contain an arithmetic in relation to 9?i. If an and a
exist such that for every 3 of there is a synonymous sub-
stitution-value of 93^ in Si and that in Si has infinite univer-
sality, then we say that S contains a general arithmetic (in relation
to %).
3i and 32 called corresponding 3 in 9?i and 9?2 if n exists
such that 3i is an «-3 of 9li, and 32 is an «-3 of 9i2- Here 9li and
9^2 niay belong to different levels, and even to different languages.
We say that two numerical Sg” (or two numerical ^r”) (in one or
two languages) have a corresponding extent if every two full sen-
tences of them with corresponding 3 ss arguments are either both
valid or both contravalid.
If S contains an arithmetic, then it certainly contains expres-
sions which can be interpreted as designations of real numbers,
namely the numerical Sg^ ; and further, it may contain numerical
and numerical of which the full expressions are 3 (see
§ 39)- ^0 will call 95i a 93 for real numbers if there are infinitely
many numerical (or numerical of the kind mentioned)
which belong to the substitution-values of 95i. If 93i is a 93 for real
numbers and if 93i in Si has infinite universality, then we call Si
a universal sentence concerning real numbers. The arithmetical
equality between two real numbers that are represented by tw^o
Sg^ (or two ^r^) in relation to the same 3“Series 91i finds its
syntactical expression in the coextensiveness of the two Sg (or
^iPr, respectively), or in the case of gu in the equality of the course
of values. If, however, it is a question of different 9li and 5R2 which
may also belong to different languages, the arithmetical equality
207
§58. ARITHMETIC
will be represented by correspondence in extent, in this way, real
numbers of various languages can be compared with one another ;
an expression can be characterized as being the expression of a
particular real number (for example : ‘ 7r-expression in relation to
9?!*). We can easily see how it may be syntactically determined
whether a dijferential and integral calculus and a theory of functions
of more or less wide extent is contained in S. We shall not go any
further into this question here.
Examples: i. Language /. The following series are examples of
3-series. %: ‘0*, ‘OH*, %: ‘O’, ‘ON*, ‘ONN’,...; ‘3’,
‘3'^...; ‘O’, ‘nf(0)’, ‘nf(nf(0))’, . . ; %: ‘3’, ‘fak(3)’,
‘fak(fak(3))* The fu of I are jfu in relation to each of these
series ; and moreover ‘ • ’ is also a 3f u in relation to each of these
scries, and specifically it is the series-forming 5fu in 91i. ‘sum’ is a
sum-f u, ‘ prod * a product-f u. Language I contains a general arith-.
metic inasmuch as there are sentences with free 3 in it. Real num-
bers can be represented in Language I by means of pr' or f ; there
is, however, no 33 for real numbers and no "ipr for rca^-number
arguments.
2. Language II (see § 39). Here also, the aforesaid series
. . . 9I5 are 3“Series ; but there are also others of quite different
kinds. The *pr can be used as pr of real numbers. Since there are
^p, ^p, and having infinite universality, there are consequently
universal sentences concerning real numbers and functions of real
numbers, etc.
§ 59. The Non-Contradictoriness and
Completeness of a Language
S is called contradictory (or d -inconsistent) if every sentence of
S is demonstrable ; otherwise, non-contradictory (or d-consist-
ent). [It is to be noted that the term ‘ contradictory * when applied
to sentences (German : kontradiktorisch) is an L-c-term (see § 52),
but when applied to languages (German: widerspruchsvoll) is a
d-term and not an L-term.] The following c-terms correspond
to these d-terms. S is called inconsistent if every sentence of S is
valid ; otherwise, consistent. If the L-sub-language of S is contra-
dictory (or non-contradictory, inconsistent or consistent), then S
is called L-contradictory (or L-non-contradictory, etc., re-
spectively). The relations between the defined d-, c-, and L-terms
are indicated by the arrows shown in the table on p. 210.
Theorem 59.1. If S is contradictory (or inconsistent), then every
2o8
PART IV. GENERAL SYNTAX
it and every S is, at the same time, both demonstrable and re-
futable (or valid and contravalid, respectively); there are no
irresoluble (or indeterminate) it or S.
Theorem 59.2. If, in S, there is a it or an S which is either
non-demonstrable (or non-valid) or non-refutable (or non-
contravalid, respectively), then S is non-contradictory (or con-
sistent, respectively). By Theorem i.
Theorem 59.3. If, in S, there is a it or an 6 which is at the same
time both demonstrable and refutable (or valid and contravalid),
then S is contradictory (or inconsistent, respectively); and con-
versely.
Theorem 59.4. If S contains the ordinary sentential calculus
(with the negation ‘ ^ *) then in S every sentence is derivable from
01 and Si- In I and II this is arrived at with the help of PSI i
and PSII I, respectively.
Theorem 59.5. If S contains the ordinary sentential calculus,
then S is contradictory when and only when an 0 i exists such that
01 and ^ 01 are demonstrable in S. By Theorem 4.
Theorem 59.6« If S contains a negation 93 Ii, then S is incon-
sistent when and only when an 0i exists such that 0i and 93 {i (0i)
are valid in S. By Theorem 57.1.
The definitions of ‘contradictory’ and ‘non-contradictory*
correspond (as Theorem 5 shows) to the ordinary use of language
without, however, negation being assumed. (See Tarski
[Methodologie] I, p. 27 f., and Post [Introduction],)
A non-contradictory language may nevertheless be inconsistent.
For although it contains no d-contradiction, it may still contain
a c-contradiction, that is to say, a contradiction which depends
upon the c-rules only. This is the reason for introducing the
narrower term ‘ consistent which applies only to languages that
contain no contradictions of any sort.
Theorem 59.7. If S is inconsistent or contradictory, then it is
true that : {a) every two sentences of S are equipollent ; {h) every
two expressions of S which are isogenous are synonymous.
Theorem 59.8. If S is inconsistent or contradictory, then S con-
tains no 3 -series and therefore no arithmetic. — By Theorem ']b\
different terms of a 3-series are not synonymous.
Example of a non-contradictory but inconsistent language, het SfUi
be ‘[(jc>0).(y>0).(«>0).(M>2)]D(x“+y“ + 2“)*. Let Si be
§ 59 - non-contradictoriness and completeness 209
()(SfUi), in other words, Fermat’s theorem. Let every closed
logical variant of SfUi be demonstrable in S (hence, for every indi-
vidual set of four positive integers Fermat’s property can be demon-
strated). Let 61 itself be analytic but non-demonstrahle ; i.e. let S
contain an indefinite rule analogous to DC 2 (p. 38) by which Si
is a direct consequence of the class of those variants. Further, let
the sentence Si (although possibly contradictory in classical
mathematics) be demonstrable in S (for instance, laid down as a
primitive sentence, other sentences such as are d-incompatible with
it being cancelled). Then S is inconsistent (and moreover L-incon-
sistent). At the same time, however, S may be non-contradictory,
since Si and Si are not both demonstrable. There is indeed no
d-contradiction here but there is a c-cortradiction — namely, that
between the class of those variants and Si. This c-contradiction
is evident in the ordinary material interpretation : the demonstrable
sentence Si means that not all sets of four positive integers have
Fermat’s property, while for every such set a demonstrable variant
occurs, which means that this quadruple has Fermat’s property.
But the c-contradiction, the inconsistency, is also purely formal
without any reference to material interpretation: the class which
consists of those variants and ^ Si contains only demonstrable
sentences but is nevertheless contradictory, that is to say, every sen-
tence is a consequence of it ; hence every sentence of S is at the same
time both analytic and contradictory.
For such languages as have no other D than the 3, our term ‘ con-
sistent’ corresponds to Godel’s term, [Unentscheidbare] p. 187, ‘a>~
non-contradictory*; see also Tarski [Widerspruchsfr.].
The language S is called complete (or d-complete) if the sen-
tential null-class (and hence, according to Theorem 48.8, every R)
is complete (or d-complete, respectively); otherwise, incomplete
(or d-incomplete). The language S is called determinate (or
resoluble) if every R (and hence every 0 also) is determinate (or
resoluble, respectively) in S; otherwise, indeterminate (or irre-
soluble). The corresponding L-teriris (‘ L-complete * and so on) are
only attributed to the language S when the original term (‘com-
plete ’, etc.) is attributable to the L-sub-language of S.
Theorem 59.9« If S is complete, then it is determinate ; and
conversely. By Theorem 48.5.
Theorem 59.10. If S is complete, then it is logical ; and con-
versely. By Theorem 50.2 a.
Theorem 59.11. If S is complete, then it is L-coinplete; and
conversely. By Theorems 10 and 51. i.
Theorem 59.12. {a) The terms ‘ complete language *, ‘ L-com-
plete language’, ‘determinate language’, ‘logical language’ co-
210
PART IV. GENERAL SYNTAX
incide. {b) The terms ‘ incomplete language *, ‘ L-incomplete lan-
guage indeterminate language V^^criptive language* coincide.
Theorem 59.I3. If S is d-complete, then it is resoluble; and
conversely. By Theorem 48.5.
For the d-terms, no valid theorems analogous to Theorems 1 1
and 12 exist.
Theorem 59.14. (a) If S is contradictory, then S is both d-
complete and complete. (^) If S is inconsistent, then S is com-
plete. By Theorem i .
How the properties of languages here defined are transferred
from one language to another can be seen from the table on
p. 225 (B). The relation of the terms to one another is indicated
by the arrows in the table below (as on p. 183).
Properties of languages
L-d-terms :
d-terms :
L-contradictory
contradictory
L-non-contra-
non-contra-
dictory
dictory
L-d-complete 1
L-resoluble J
1 / d-complete ^
^ \ resoluble J
L-d-incomplete^
L-irresoluble j
1 / d-incomplete^
f \ irresoluble j
c-terms: L-c-terms:
inconsistent <- L-inconsistent
consistent -> L-consistent
complete \ fL-complete
determinate L-determinate
logical J
f incomplete \ /L-incomplete
indeter- f ^ | (L-indeterminai
minate ^ synthetic
descriptive J
We shall see that every consistent language which contains a
general arithmetic h irresoluble. Only poorer languages are re-
soluble, for example, the sentential calculus. A richer language,
though not resoluble, can yet be determinate and complete, pro-
vided that sufficient indefinite rules of transformation are laid down.
This is the case, for instance, with the logical sub-languages of I
and II. For such an irresoluble but complete language, the following
classification of sentences holds ; it is at the same time the classi-
fication of the logical sentences of any irresoluble language what-
soever (for the classification of the descriptive sentences, seep. 185) :
(d-terms :)
demonstrable
A
irresoluble refutable
r
1
V Y ■ ''
1 1
1
\ V"—
1
A
(c- and L-
terms :)
valid
analytic
Y
contravalid
contradictory
6oa. THE ANTINOMIES
2II
§ 6oa. The Antinomies
In investigating the non-contradictoriness of a language, the
first thing to be asked is whether the familiar so-called anti-
nomies or paradoxes which appeared in earlier systems of logic and
of the Theory of Aggregates have definitely been eliminated. This
point is an especially critical one when we are concerned with a
language which is rich enough to formulate, to any extent, its own
syntax, whether in an arithmetized form or with the help of special
syntactical designations. The syntactical sentences may sometimes
speak about themselves, and the question arises whether this re-
flexiveness may not possibly lead to contradictions. This question
is significant because it is not concerned with calculi of a specially
constructed kind but yrith all systems whatsoever which contairi
arithmetic. We shall now investigate this question and in doing so
we shall avail ourselves of the results obtained by GodeJ.
We shall follow Ramsey’s example in dividing the antinomies
into two kinds, and we shall see that those of the second kind are
the ones which come into consideration for our inquiry. These will
therefore be examined more closely. In the examples we propose
to use partly the word-language and partly a symbolism similar to
that which was used in Language II ; for the syntactical designa-
tions we shall employ in some cases Gothic symbols, and in others
inclusion in inverted commas. Let us consider, to begin with, the
following two antinomies.
1. Russell’s antinomy [Princ. Math, I] ; [Math. Phil.]. We de-
fine as follows : a property is called impredicahle when it does not
apply to itself. Expressed in symbols: “Impr(F) = If
in this case we substitute ‘ Impr’ itself for ‘F*, we get the contra-
dictory sentence : “ Impr (Impr) = Impr (Impr)**.
2. Grelling’s antinomy. Definition : in a language which con-
tains its own syntax, a syntactical predicate (for example, an
adjective) is called heterological if the sentence which ascribes the
property expressed by the predicate to the predicate itself is false.
If, for instance, ‘ Q ’ is a syntactical predicate, then “ Het (‘ Q ’) =
is true. (The fundamental difference between this
antinomy and the foregoing, which is disregarded in many pre-
sentations, is to be noted, namely, that here the property Q is
attributed, not to the property Q but to the predicate, i.e. the
212
PART IV. GENERAL SYNTAX
symbol * Q *.) Example: the adjective ‘ monosyllabic ’ is heterological
because ‘monosyllabic* is not monosyllabic but penta-syllabic.
Now, if instead of the predicate ‘ Q we take the predicate ‘ Het *
itself which has just been defined, we get, from the definition as
stated, the contradictory sentence “ Het (‘ Het *) = '^ Het (‘ Het ’) **.
In order to avoid antinomies in his language, Russell set up a
complicated rule of types, which, particularly in the theory of real
numbers, gave rise to certain difficulties, to overcome which he
found it necessary to state a special axiom, the so-called Axiom of
Reducibility. Ramsey {[Foundations] Treatise i, 1925) has shown
that the same object may be attained by a far simpler method. He
discovered, namely, that it is possible to differentiate between two
kinds of antinomies which may be designated as logical (in the
narrower sense) and syntactical (the latter are also called linguistic,
epistemological, or semantic). Example (i) belongs to the first
category and (2) to the second. Following Peano, Ramsey pointed
out that the antinomies of the second kind do not appear directly
in the symbolic system of logic, but only in the accompanying
text; for they are concerned with the expressions. From this fact
he drew the practical conclusion that in the construction of a
symbolic system it is not necessary to take note of these syntactical
antinomies. Now since the antinomies of the first kind are already
eliminated by the so-called simple rule of types, this is sufficient ;
the branched rule of types and the axiom of reducibility which it
necessitates are superfluous.
On the basis of the simple rule of types (as in II for instance) the
type of a predicate is determined by the type of the appertaining
arguments alone. On the basis of Russell’s branched rule of types,
the form of the chain of definitions of a predicate is also a factor
in determining its type (for instance, whether it is definite or not).
But the simple rule of types is sufficient to determine that a predicate
always belongs to a type other than that of the appertaining argu-
ments (namely, that it always belongs to a type of a higher level).
Thus, here, a sentence cannot have the form ^F{F)\ And hence a
definition of the form given for ‘ impredicable * is obviously impos-
sible. In the same way, the other well-known antinomies of the first
kind are obviated by means of the simple rule of types.
The problem of the syntactical antinomies^ however, obviously
reappears when it is a question of a language S in which the syntax
of S itself can be formulated, and therefore in the case of every
213
§ 60 fl. THE ANTINOMIES
language which contains arithmetic. There is a prevalent fear that
with a syntax of this sort, which refers to itself, either contradic-
tions similar to the syntactical antinomies will be unavoidable, or
in order to avoid them, special restrictions, something like the
“ branched ” rule of types, will be necessary. A closer investigation
will show, however, that this fear is not justified.
The above-mentioned view is held, for instance, by Chwistek.
He had already, before Ramsey, had the idea of stating only the
simple rule of types, and thus rendering the axiom of reducibility
unnecessary. Later, however, he came to the conclusion that with
the rejection of the branched rule of types the syntactical anti-
nomies — that of Richard, for example — ^would appear (see Chwistek
[Nqm. Ot-undl.]). In my opinion, however, the indispensability of
the branched rule of types in Chwistek’s system is due only to the
fact that he uses the autonymous mode of speech for his syntax (the
so-called Semantics) (see § 68). •
Apart from Grelling’s, the most important example of a syn-
tactical antinomy is the one which was already famous in ’antiquity,
the antinomy of the liar (for the history of this see Riistow).
Someone says : I am lying or more exactly : “ I am lying in this
sentence”, in other words: “This sentence is false.” If the sen-
tence is true, then it is false ; and if it is false, then it is true.
Another antinomy which belongs to the category of the syn-
tactical antinomies is Richard’s (see [Princ, Math,] I, 6i, and
Fraenkel [Mengenlehre] p. 214 ff.). In its original version it is con-
cerned with the decimals definable in a particular word-language.
It can be easily transferred to 3pr^ in the following manner. Let S
be a language whose syntax is formulated in S. In S there are at
most a denumerable number of 3pr which are definable. Therefore
we can correlate univocally a natural number with every such 3pr^
(for instance, by a lexicographical arrangement of the definition-
sentences or, in an arithmetized syntax, simply by the term-
number of the 3pr^). Let ‘ c ’ be a numerical expression ; we will
call the number c a Richardian number if c is the number of a
3pr^, say ‘P’, which does not appertain to the number c, so that
‘P(c)* is false (contradictory). Accordingly, the adjective
* Richardian’ is a defined jpr^, and thus has correlated with it a
certain number, say b- Now b must be either Richardian or not.
If b is Richardian, then, according to the definition, the property
having the number b does not appertain to b; therefore, in this
214
PART IV. GENERAL SYNTAX
case, in contradiction to our assumption, b is not Richardian.
Hence b must be non -Richardian. b must leave the definition of
‘ Richardian * unfulfilled, and therefore must possess the property
having the number b ; that is to say, b must be Richardian. This is
a contradiction. ^
It is characteristic of the syntactical antinomies mentioned that
they operate with the concepts ‘ true * and ‘ false *. For this reason
we will examine these concepts more closely before considering the
syntactical antinomies any further.
§ 606. The Concepts ' True ' and ‘ False '
The concepts ‘true* and ‘false* are usually regarded as the
principal concepts of logic. In the ordinary word-languages, they
are used in such a way that the sentences ‘ Si is true * and ‘ Si is
false’ belong to the same language as Si. This customary usage of
the terms ‘ true ’ and 'false ’ leads^ however y to a contradiction. This
will be shown in connection with the antinomy of the liar. In order
to guard ourselves against false inferences, we will proceed in a
strictly formal manner. Let the syntax of S formulated in S con-
tain three syntactical adjectives, ‘91’, ‘3EB*, ‘ 5*, concerning which
we will make only the following assumptions (V 1 -- 3 ). In these,
we shall write the sentence : “ 9 li has the property 91 ” in an abbrevi-
ated form, thus: ‘9l(9Ii)’. If ‘9l(9Ii)’ is interpreted as “9Ii is a
non-sentence ”, ‘ 3B (9li) ’ as : ‘‘ The expression 9Ii is a sentence, and,
specifically, a true sentence”, and ‘ 3f(9li)’ as: “9li is a sentence,
and, specifically, a false sentence”, then our assumptions V 1-3
are in agreement with the ordinary use of language.
V I . Every expression of S has exactly one of the three properties
91, SB, 5 .
\ 2 a. Let ‘ A ’ be any expression whatsoever of S (not : “ desig-
nation of an expression”); if SB (‘A’), then A. [For instance: if
“this tree is high” is true, then this tree is high.]
Vzh. If A, then SB (‘A’).
V 3 . For any SIi, the expressions ‘9l(Sli)*, ‘SB(9Ii)*, ‘5(Sli)’
do not possess the property 91 (hence, they do possess either SB
or according to V i).
From V i and 2 5 it follows that:
If 5 (‘A’), thqn not SB (‘A’), and therefore not A. ( 4 )
215
§ bob, THE CONCEPTS ‘ TRUE * AND ‘ FALSE *
From V i and 2 a it follows that:
If not A, then not2B(‘A*), and therefore 5 (‘A*), or^(‘A*). ( 5 )
Now in analogy with the assertion of the liar, it is easy to show
that the investigation of an expression ?I 2 with the text ‘ g (^tg) * leads
to a contradiction. The fact that an expression is here designated
by a symbol (namely: ^312*), which itself occurs in itself, easily has
a confusing effect. But we can also establish the contradiction
without this direct reflexive relation; it is not, as is so often be-
lieved, the reflexiveness which constitutes the error upon which
the contradiction depends ; the error lies rather in the unrestricted
use of the terms ‘ true * and ‘ false *. Let us examine the two ex-
pressions ‘ g (9li) * and ‘ 2B (?l 2 ) *. Obviously these are expressions,
at worst non-sentences. We are entirely at liberty as to which ex-
pressions we choose to designate by and let us agree
that:
(a) SIj shall be the expression ‘ 9 B( 3 l 2 )’; {h) shall be the ex-
pression
(Here, as can be seen, no designation of an expression occurs in
the expression itself.)
According to V 3 :
Either m C 5 W) or 5 (* (?)
We first make the assumption: 3B (‘ g(3li)’). From this, in
accordance with V 2 a, it would follow that: g(‘ili). This, ac-
cording to ( 6 a) is 3 f (‘ 3 B(?l 2 )*); from which, according to ( 4 ),
would follow: not 2 B( 9 l 2 ). This is, by {6b): not 2B (‘ g (9li) ’ ).
Our assumption leads to its own opposite and is therefore refuted.
Thence, according to ( 7 ), it is true that :
5CafW’)-
( 8 )
From this, by ( 4 ), follows :
not
(9)
This, according to (6 a) is :
not 3f (‘28(31,)’).
(lo)
ByVs:
'
SB (‘38(34)’) or 5 (‘28(34)’).
(n)
From (lo) and (ii):
28 (‘28(31,)’).
(l 2 )
2i6
PART IV. GENERAL SYNTAX
Thence, in accordance with V za:
JBW-
( 13 )
From (8) and (6 b ) :
tSfW-
(14)
Therefore, in accordance with V i :
notSBW.
(IS)
(13) and (15) constitute a contradiction.
This contradiction only arises when the predicates ‘true’ and
* false * referring to sentences in a language S are used in S itself.
On the other hand, it is possible to proceed without incurring any
contradiction by employing the predicates ‘true (in Si)* and ‘false
(in Si) * in a syntax of Sj which is not formulated in Si itself but in
another language Sg. Sg can, for instance, be obtained from Sj by
the addition of those two predicates as new primitive symbols and
the erection of suitable primitive sentences relating to them, in the
following way: i. Every sentence of Si is either true or false.
2. No sentence of Sj is at the same time both true and false.
3. If, in Sj, S 2 is a consequence of 5li, and if all sentences of 5li are
true, then Sj likewise true. A theory of this kind formulated in
the manner of a syntax would nevertheless not be a genuine syntax.
For truth and falsehood are not proper syntactical properties; whether
a sentence is true or false cannot generally be seen by its design, that
is to say, by the kinds and serial order of its symbols. [This fact
has usually been gverlooked by logicians, because, for the most
part, they have been dealing not with descriptive but only with
logical languages, and in relation to these, certainly, ‘true* and
‘false* coincide with ‘anal)rtic* and ‘contradictory*, respectively,
and are thus syntactical terms.]
Even though ‘true* and ‘false* do not in general occur in a
proper syntax (that is to say, in a syntax which is limited to the
design-properties of sentences), yet the majority of ordinary sen-
tences which make use of these words can be translated either into
the object-language or into the syntax- language. If Si is ‘ A *, then
‘Si is true* can, for example, be translated by ‘A*. In logical in-
vestigation, ‘ true * (and ‘ false *) appears in two different modes of
use. If the truth of the sentence in question follows from the rules
of transformation of the language in question, then ‘ true * can be
translated by ‘valid* (or, more specifically, by ‘analytic*, ‘de-
217
§6o6. THE CONCEPTS ‘true* AND ‘false*
monstrable *) and, correspondingly, ‘false* by ‘contravalid’ (or
‘contradictory*, ‘refutable*). ‘True* may also refer to indeter-
minate sentences, but in logical investigations this only happens in
the conditional form, as, for example ; ‘ If 0 ^ is true, then Sg true
(or false, respectively).^* A sentence of this kind can be translated
into the syntactical sentence : ‘ S2 is a consequence of Sj (or is in-
compatible with Si, respectively).*
§ 60c. The Syntactical Antinomies
We will now return to the question whether, in the formulation
of the syntax of S in S, contradictions of the kind known as
syntactical antinomies may not arise if, in the ordinary phrasing
of these antinomies, ‘ true * and ‘ false * are replaced by syntactical
terms in the manner indicated above.
Let S be a non-contradictory language (and, further, a con-
sistent one), which contains arithmetic, and hence an arrthmetized
syntax of S itself also. Then a certain method exists whereby it is
possible to construct, for any and every syntactical property
formulable in S, a sentence of S, Si, such that Si attributes this
property — ^whether rightly or wrongly — to itself. This has already
been shown in the case of Language II (see § 35). Now, by means
of a construction of this kind, we will try to restate the antinomy
of the liar. It consists of a sentence which asserts its own false-
hood.
First, let us replace ‘false* in this antinomy by 'non-demon-
sir able *. If we construct a sentence of S, Si, which asserts of itself
that it is non-demonstrable in S, then we have in Si an analogue
to the sentence (5 of Language II which has already been dis-
cussed (and to the sentence ©j of Language I). Here no contra-
diction arises. If Si is true (analytic), then Si is not false (contra-
dictory), but is only non-demonstrable in S. This is actually the
case (see Theorem 36.2). The properties ‘analytic* and ‘non-
demonstrable * are not incompatible.
Now let us replace ‘ false * by ‘ refutable * in the sentence of the
liar. Assume that a sentence, S2, is constructed in S which asserts
that S2 is itself refutable (in S). S2 is then an analogue to the
assertion of the liar. We will now observe whether the contradic-
tion arises in the ordinary way. First let us assume that S2 is
2i8
PART IV. GENERAL SYNTAX
actually refutable. Then S2 will be true, and therefore analytic.
On the other hand, however, every refutable sentence is contra-
dictory, and hence not analytic. Therefore the assumption is a
false one and S2 non-refutable. From this no contradiction
follows. S2 is actually non-refutable ; since ©2 means the opposite
of this, 02 false, and is therefore contradictory. But the pro-
perties ‘non-refutable* and ‘contradictory* are quite consistent
with one another (see the diagram on p. 210) ; for instance ‘ '^ ( ) ( 05 )
possesses both.
The impossibility of reconstructing the antinomy of the liar
with the help of the terms ‘ non-demonstrable * or ‘ refutable * is due
to the fact that not all analytic sentences are also demonstrable, and
similarly not all contradictory sentences are also refutable. But
what would happen if we were to use in place of ‘ true * and ‘ false *
the syntactical terms ‘ analytic ’ and ‘ contradictory * ? Like ‘ true *
and ‘ false *, these two terms constitute a complete classification of
the logical sentences. It is easy to show that we can construct
contradictions if we assume that ‘analytic (in S)* and ‘contra-
dictory (in S) * are defined in a syntax which is itself formulated in
S. We could then, of course, construct a logical sentence Sg which,
in material interpretation, would mean that ©3 was contradictory.
03 would correspond exactly to the assertion of the liar. Since it
would be a logical sentence, 03 would be either analytic or contra-
dictory. Now, if 03 were contradictory, ©3 would be true, there-
fore analytic, therefore not contradictory. Hence, 03 would have
to be non-contradictory. But then 03 would be false, and there-
fore contradictory — which would be a contradiction.
On the same assumptions it would be possible also to construct
Grelling's antinomy. Let us state the procedure for Language II.
Assuming that a predicate ‘An* is definable in II in such a way
that ‘An(jc)’ means: “The SNgentence x is analytic (in II).**
‘ Heterological* could then be defined as follows: ‘Het(jc) =
^ An (subst [jc, 3, str (jc)]) *. Let ‘ Het (x) * have the series-number b.
Then it is easy to show that, for the sentence ‘Het(b)*, either
assumption — that it is analytical or that it is contradictory — leads
to a contradiction.
We have seen that if ‘ analytic in S * is definable in S, then S
contains a contradiction; therefore we arrive at the following
result:
219
§ 6 oc. THE SYNTACTICAL ANTINOMIES
Theorem 6oc.i. If S is consistent, or, at least, non-contra-
dictory, then * analytic {in S)* is indefinable in S. The same thing
holds for the remaining c-terms which were defined earlier (in so
far as they do not coincide with d-terms), for instance, ‘ valid *,
‘consequence*, ‘equipollent*, etc. But it is not true for every
c-term which does not coincide with a d-term.
If a syntax of a language Si is to contain the term ‘analytic
(in Sj) * then it must, consequently, be formulated in a language Sg
which is richer in modes of expression than Sj. On the other hand,
the d-term ‘ demonstrable (in Si) ’ can, under certain circumstances,
be defined in Si ; whether that is possible or not depends upon the
wealth of modes of expression which is available in Si- With
Languages I and II the situation on this point is as follows:
‘analytic in I* is not definable in 1 , but it is definable in II;
‘ analytic in II * is not definable in II, but is only definable in a still
richer language. * Demonstrable in I *, because it is indefinite, is
not definable in I; but ‘demonstrable in II* can be defined in II,
namely, by means of ‘ (3 r) [BewSatzII (r, x)] *.
The foregoing reflections follow the general lines of Coders
treatise. They show also why it is impossible to prove the non*
contradictoriness of S in S. Closely related to Theorem i is the
following theorem (a generalization of Theorem 36.7 ; see Godel
[Unentsekeidbare], p. 196; Godel intends to give a proof of this
generalized theorem in a continuation of that treatise).
Theorem 6oc.2. If S is consistent, or at least non-contradictory,
then no proof of the non-contradictoriness or consistency of S can be
formulated in a syntax which uses only the means of expression which
are available in S.
The investigation of Richard’s antinomy (p. 213) leads to a
similar conclusion. Assume that in S there is an by means of
which a univocal enumeration of all the 3pr^ which are definable
in S might be constructed. This could be effected, for example, by
means of an fui such that every full expression fUi(3pr^) was a 3-
We will use the symbolism of II and write fUi ‘ num *.
The univocality of the numbering is assumed ;
(num (F) = num (G) ) D (x) (F (jc) = G{x)j. ( i )
With the help of ‘ num *, ‘ Ri * (“ Richardian **) could now be defined :
Ri(A:) = (F) [(num(F) = a:)D^F(x)]. (2)
220
PART IV. GENERAL SYNTAX
Since * Ri* is a it has a certain particular number designated
by ‘num(Ri)*. We assume first that the number of ‘Ri* is itself
Richardian: * Ri [num (Ri)] *. Then if we substitute in (2)
‘num(Ri)* for and ‘Ri’ for ‘F’, ‘ ^ Ri [num (Ri)] ’ easily
follows. Since our assumption leads to its opposite, it follows that
it is refuted ; and therefore it is proved that
/--'Ri[num(Ri)]. (3)
From (i):
(num (F) = num (Ri) ) 3 ('^F [num (Ri)] = Ri [num (Ri)] ).
( 4 )
From (3), (4):
(num (F) = num (Ri)) D ^F [num (Ri)]. (S)
From (2) :
(F) [(num(F) = num(Ri))D^F[num(Ri)]]D Ri [num(Ri)].
( 6 )
From (5), (6):
Ri[num(Ri)]. (7)
The proved sentences (3) and (7) contradict one another; S is
therefore contradictory. Thence follows:
Theorem 6oc.3. If S is consistent, or at least non-contradictory,
then it is not possible to construct in S either an or an by
means of which a univocal enumeration of the 3pr' of S could be
constructed. — Although the aggregate of the 3pr^ which are de-
finable in S is a .denumerable aggregate, in accordance with this
Theorem an enumeration of them cannot be effected with the
means available in S itself. [The condition in this Theorem is only
added for the purpose of facilitating understanding; if S is in-
consistent, then in S no univocal enumeration of number of
objects is possible at all, since no (non-synonymous) 3
available.]
§ 6od. Every Arithmetic is Defective
Let Si contain an arithmetic (in relation to a certain 3 '‘Series),
and let the real numbers be represented in Sj by 3fu^ Let Sj be
a conservative sub-language of Sg, and let the arithmetized syntax
of Si be formulated in Sg. We will show that with the help of the
arithmetico-syntactical terms of Sg, as referred to Si, a can be
221
§ bod. EVERY ARITHMETIC IS DEFECTIVE
defined in Sg for which there is no 3fui in Sj having the same course
of values ; this is true for every language Sj, however rich it may be,
if we take a sufficiently rich language as Sg. We define the 3fu^
‘k’ in Sg in the following way: i. If ac is not a term-number of a
3fu^ of Si, then k(A£r) =» 0 ; 2. If jc is a term-number of a 3fu' of Sj,
let us say ‘ h ’, then k (At) = h (jc) + i. Then every 3fu' of Sj deviates
from * k * for a certain argument (namely, for its own term-number) ;
and therefore in Sj there is no 3fu^ having the same course of
values as * k *. In other words : a real number can be given which is
not equal to any real number definable in Sj (see p. 206).
Theorem 6od.i« For every language S a real number which
cannot be defined in S can be given.
The above definition of ‘k* corresponds to the so-called
diagonal method of the Theory of Aggregates. Theorem i coin'e^
spends to the well-known theorem of the Theory of Aggregates
which states that the aggregate of the real numbers is a non-
denumerable aggregate. (On the concept of the non-denumerable
aggregates see, however, § 71 rf.) On the other hand, the above
line of thought also corresponds to Richard^s antinomy.
We will now summarise briefly the results of this investigation
of the syntactical antinomies. Let the syntax of a language S be
formulated in S. The reconstruction of the syntactical antinomies
by means of terms which are defined in S (for instance, in Lan-
guage II, ‘ non -demonstrable in IL or ‘ refutable in II ') does not
lead to contradictions; but it opens the way to the proof that
certain sentences are non-demonstrable or irresoluble in S. With
the help of other terms (for instance, ‘ analytic *, ‘ contradictory ’,
‘consequence’, ‘correlated number’, ‘term-number’) the recon-
struction of the syntactical antinomies is possible. This leads to the
proof that these terms (of which the definitions have up to now
only been formulated in words and not within a formalized system)
cannot be defined in S, if S is consistent, or at least non-contra-
dictory. Since terms and sentences of pure syntax are nothing
other than syntactically interpreted terms and sentences of arith-
metic, the investigation of the syntactical antinomies leads to the
conclusion that every arithmetic which is to any extent formulated
in any language is necessarily defective in two respects.
Theorem 6od.2. For every arithmetical system it is possible to
state: {a) indefinable arithmetical terms and (6) irresoluble arith-
222
PART IV. GENERAL SYNTAX
metical sentences (Godel [Unentscheidbarel^. In connection with
(a) see Theorems 6 oc.i, 3 , 6od.i. In connection with (b) see
Theorem 60 C. 2 ; further irresoiuble sentences analogous to (5 in
II and (Sj in I (see § 36 ) can be constructed.
This defectiveness is not to be understood as it there were, for
instance, arithmetical terms which could not be formally (i.e. in a
calculus) defined at all, or arithmetical sentences which could not
be resolved at all. For every term which is stated in any un-
ambiguous way in a word-language, there exists a formal defini-
tion in an appropriate language. Every arithmetical sentence Si
which is, for instance, irresoiuble in the language Si is yet de-
terminate in Si; in the first place there exists a richer syntax-
language Sg, within which the proof either that Si is analytic or
tha.t Si is contradictory can be stated ; and secondly, there exists
an object-language S 3 of which Si is a proper sub-language, such
that Si is resoluble in S3. But there exists neither a language in
which all arithmetical terms can be defined nor one in which all
arithmetical sentences are resoluble. [This is the kernel of truth
in the assertion made by Brouwer [Sprache]^ and, following him,
by Heyting [Logik], p. 3 , that mathematics cannot be completely
formalized.] In other words, everything mathematical can be
^ormalizedy but mathematics cannot be exhausted by one system \ it
requires an infinite series of ever richer languages.
(d) TRANSLATION AND INTERPRETATION
§ 61. Translation from One Language
INTO Another
We call Qi a syntactical correlation between the syntactical ob-
jects (?l or 51) of one kind and those of another when Qi is a many-
one relation by means of which exactly one object of the second
kind is correlated to every object of the first, and every object of
the second kind to at least one of the first. The 91 (or Si) which is
correlated to 9li (or 5^i, respectively) by- means of Qi is called the
SOi^-correlate of 9(i (or of and is designated by ‘Oi[9Ii]* (or
Herein the following condition is assumed: if 9In has
no direct Qj^-correlate but can be subdivided into the expressions
91i,9l2, ...9I,n. which have such correlates, then Qi[9ln] is equal
§6l. TRANSLATION FROM ONE LANGUAGE INTO ANOTHER 223
to the expression composed of Qi[ 9 li],Qi[ 9 y, The
class which contains all and only the Q^-correlates of the sen-
tences of ill is designated by ‘Qi[Sli]'. According to this, the
correlates of sentences are also determined by means of a correla-
tion between expressions, and the correlates of sentential classes
by means of a correlation between sentences. [In a formalized
syntax, Qj can, for instance, be either an a an 9tgi, or an
] We say that a certain syntactical relation is transformed into
a certain other one by means of Qj if, when the first relation sub-
sists between any two objects, the second subsists between the
Qi-correlates of these objects.
A syntactical correlation, Qj, between all sentential classes (or
all sentences, or the expressions of an expressional class 5li, or all
symbols) of Si and those of S2, is called a transformance of Si into
S2 in respect of classes (or of sentences, or expressions, or symbols,
respectively) provided that, by means of Qi, the consequence rela-
tion in Si is transformed into the consequence relation in Sg. For
ill assumed that no expression of ilj, but every sentence of Sj
which does not belong to ilj, is univocally analyzable into several
expressions of ili. Qi is called a transformance of Si into Sg if Qi is a
transformance of Si into Sg of one of the kinds mentioned. * h-trans-
formance in respect of classes (sentences, and so on) * is analogously
defined, the requirement in this case being the maintenance of
the relation ‘ L-consequence \
Theorem 6i.i« If Qi is a transformance of Si into Sg, then Oi
is also an L-transformance of Si into Sg.
Theorem 61.2. If JQi is a transformance of Si into Sg in respect of
sentences, then by Qi the consequence relation between sentences
in Si is transformed into the consequence relation between sen-
tences in Sg. The converse is not universally true.
A transformance of Si into Sg is called reversible when its con-
verse (that is, the relation subsisting in the reverse direction) is a
transformance of Sg into Si ; otherwise irreversible.
Theorem 6i.3* Let Qi be a transformance of Si into Sg ; if Qi is
reversible, then Qi is a one-one relation. The converse is not uni-
versally true.
Example of an irreversible transformance in respect of sentences :
the transformance given by Lewis [Logic], p. 178, of his system of
strict implication (without the existential postulate) into the ordinary
PART IV. GENERAL SYNTAX
224
sentential calculus. In this case, the correlate of the three sen-
tences of the first system, ‘A*, ‘M(A)', and ‘ (writing
‘ M * instead of the symbol of possibility) is the same sentence, ‘ A \
The transformance is thus not a one-one relation and is therefore
irreversible.
If there exists a transformance (in respect of classes, etc.) of
in S2, then Si is called transformable (in respect of classes, etc.) in
Sg. If Sj is reversibly transformable in Sg in respect of symbols,
then Si and Sg are called isomorphic.
Theorem 61.4. If in Sg there is a valid sentence 61 and a contra-
valid sentence Sg, then any language Si is transformable into Sg in
respect of sentences. — It is possible, for example, to take Sg as the
correlate of every contravalid sentence of Si, and Si as the corre-
late of every other sentence. This Theorem shows the compre-
hensiveness of the concept of transformability ; the concept of
reversible transformability is a much more restricted one, and
that of isomorphism more restricted still.
Theorem 61 ,5. Let Si and Sg be isomorphic with respect to the
correlation Qi- If Oi is a t)l" in Si with a characteristic, then
Qi [aj is a n!” in Sg with the same characteristic. If, for instance,
Qi is a symbol of proper negation (or of disjunction, and so on),
then Qi [dj is likewise a symbol of proper negation, etc.
Let Oi be a transformance (in respect of classes, etc.) of Si into
Sg; and let Sg be a sub-language of S3 (see diagram). Then Qi is
called a translation (in respect of classes, etc.) of Si into S3; and
Si is called translatable (in respect of classes, etc.) into S3. The fol-
lowing table gives for a number of syntactical properties and rela-
tions of ft or S (column i A), and for a number of properties of
languages (column i B), the conditions (sufficient but not neces-
sary) under which they are transferred from one to another of the
three languages between which the given relation holds: in the
direction of the transformance, that is, from fti to Qi [ftj (column
2) ; and conversely (6) ; from the sub-language to the total language
The property or relation occurring in column (i) w transferred
under the following conditions :
§6l. translation FROM ONE LANGUAGE INTO ANOTHER 225
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i6
226
PART IV. GENERAL SYNTAX
(3); and conversely (5); in the direction of the translation (4); and
conversely (7).
Abbreviations for the conditions :
gen. : generally, i.e. in all cases ;
L: where Qi is an h~transformance
R : where is a reversible transformance ;
c : where S2 is a conservative sub-language of S3 (see p. 179) ;
r: where Sj is a sufficiently rich sub-language of S3, namely,
a language containing either a R which is contravalid in
S3, or all the sentences of S3.
The conditions given in the table in square brackets refer to the
L-tcrm which corresponds to the term occurring in column (i).
Examples: If is valid in S|, then it is also valid in S3. If R^ is
analytic in Sg, then it is also analytic in S^, provided that Di is a re-
versible L-transformance. If S3 is inconsistent, then S2 is also in-
consistent, provided that S2 is a conservative sub-language of S3.
Since every' transformance is at the same time a translation
(namely, into an improper sub-language), the following theorems
and definitions can also be referred to transformances.
Theorem 6i.6. If Sj is translatable into Sg in respect of
symbols, then it is also translatable in respect of expressions ; if
in respect of expressions, then also in respect of sentences, and
conversely ; if in respect of sentences, then also in respect of classes.
Let Qi and Cg translations of Sj in Sg. We say that and Qg
coincide in content if, for every in Sj, [Rj] and Qg [^^i]
equipollent in Sg-
Let Sj and Sg be sub-languages of S3; and let be a translation
of Sj into Sg. If in this case, Rj and JQj [Rj] are always equipollent
in S3, we call Gj an equipollent translation in respect of S3. Ana-
logously, ‘ L-equipollent translation’ is defined by reference to
‘ L-equipollent classes Further, if Qj is a translation in respect of
symbols or expressions such that and py are always synony-
mous in S3, we call Gi a synonymous translation in respect of S3.
A synonymous translation is also an equipollent translation.
Theorem 61.7. If Sj is a conservative sub-language of Sg then
the equality of symbols represents a synonymous translation of Sj
in S2 in respect of 83.
Examples: Let I' be the sub-language of I constructed by means of
eliminating the variables. Then T is synonymously translatable into I
by means of the equality of symbols. Again, I is translatable into I' in
§6l. TRANSLATION FROM ONE LANGUAGE INTO ANOTHER 227
respect of classes, although is a proper sub-language of I. If, for
instance, Si is an open sentence of I with exactly one free variable,
3i, then the class of all sentences of the form Sj may be taken
as the correlate of { Si}. This translation is equipollent in respect of
I. There is no equipollent translation of I into T in respect of sen-
tences ; this example therefore demonstrates the importance of the
concept of translation in respect of classes.
Let Si be the intuitionist sentential calculus of Heyting \Logik\ ; and
let S3 be the ordinary sentential calculus (that of Language II, for
instance). The ordinary translation, Qi,of Si into S3 (that is to say, the
translation in which the symbol of negation is the Qi-correlate of the
symbol of negation, the symbol of disjunction the Qi-correlate of the
symbol of disjunction, and so on) is a transformance of Si into a
proper sub-language, S2, of S3. This transformance is one in respect
of symbols (if, in Si and S3, we insert all the brackets as in Language
II). S2 is a proper sub-language of S3, since, for instance, ‘p v ^ — ^p *
is not valid in S2. Nevertheless, S3 is also, conversely, translatable
into Si. Let Q2 be the converse of Qi ; and let Cia [ SJ be Q2 [ ^1]
(if 01 has the form 62, then Si itself can also be taken as Q3 [SJ).
Then £^3 is a translation of S3 into Si in respect of sentences. [This
translation was originated by Glivenko ; Godel gives another trans-
lation in connection with it {[Koll. 4], p. 34).]
On the concept of translation, see also Ajdukiewicz.
§ 62. The Interpretation of a Language
When only the formal rules of a language, for instance our
Language II, or the Latin language, are known, then, although it
is possible to answer syntactical questions concerning it — to say,
for example, whether a given sentence is valid or contravalid,
descriptive or existential, and so on — it is not possible to use it as
a language of communication, because the interpretation of the
language is lacking. There are two ways in which anyone may learn
to use a language as a language of communication: the purely
practical method which is employed in the case of quite small
children and at the Berlitz school of languages, and the method of
theoretical statements or assertions, such as is used, for instance,
in a text-book without illustrations. In the present work, by the
interpretation of a language we shall always mean the second pro-
cedure, that is, the method of explicit statements. Now, what form
will these interpretative statements take? To give an illustration,
when we wish to state what a certain Latin sentence means in English
we shall do so by equating it with another sentence which has the
228
PART IV. GENERAL SYNTAX
same meaning. Frequently the second sentence will likewise belong
to the Latin language (for example, whenever we explain a new word
by a familiar synonym) ; usually, however, it will be a sentence in
English, but it may also be a sentence in any other language, such
as French. The interpretation of the expressions of a language Sj
is thus given by means of a translation into a language Sg, the state-
ment of the translation being effected in a syntax- language S3;
and it is possible for two of these languages, or even all three, to
coincide. Sometimes, special conditions are imposed on the trans-
lation — for instance that it must depend upon a reversible trans-
formance, or that it must be equipollent in respect of a particular
language, and so on.
The interpretation of a language is a translation and therefore
something which can be formally represented \ the construction and
examination of interpretations belong to formal syntax. This
holds equally of an interpretation of, say, French in German when
what is required is not merely some kind of transformance in
respect of sentences, but, as we say, a rendering of the sense or
meaning of the French sentences. We have already seen that, in
the case of an individual language like German, the construction
of the syntax of that language means the construction of a calculus
which fulfils the condition of being in agreement with the actual
historical habits of speech of German-speaking people. And the
construction of the calculus must take place entirely within the
domain of formal syntax, although the decision as to whether the
calculus fulfils the given condition is not a logical but an historical
and empirical one, which lies outside the domain of pure syntax.
The same thing holds, analogously, for the relation between
two languages designated as translation or interpretation. The
ordinary requirement of a translation from the French into the
German language is that it be in accordance with sense or meaning
— which means simply that it must be in agreement with the his-
torically known habits of speech of French-speaking and German-
speaking people. The construction of every translation, and thus
of every so-called true-to-sense translation, also takes place within
the domain of formal syntax — although the decision as to whether a
proposed translation fulfils the given requirement ^nd can thus be
called true-to-sense is an historical, extra-syntactical one. It is
possible to proceed in such a way that the extra-syntactical re-
229
§62. THE INTERPRETATION OF A LANGUAGE
quirement is here of the same kind as in the first case, namely, is
concerned with the agreement of a syntactically constructed cal-
culus with a certain historically given language. We first stipulate,
for instance, that the French language be represented, say, by the
calculus Sj, the Gernjan language by Sg*, and, further, that the
language which consists of the French and German languages as
sub -languages be represented by the calculus S3, of which S^ and
Sg are sub-languages. Then a syntactically given translation, Q^,
of Si into Sg is true-to-sense if it is equipollent in respect of S3.
Under certain circumstances it will be required in addition that
Qi be a synonymous expressional translation in respect of S3.
Sometimes the interpretation of a language Si in relation to an
existing language Sg is given by constructing from Sg a more com-
prehensive language S3 by means of the addition of a sub-langujige
which is isomorphic or even congruent with Si. The interpretation
of a symbolic calculus — such as a mathematical calculus — on the
basis of an existing scientific language, is, in particu*lar, often
effected in this manner.
Examples: If, for instance, the system of the calculus of vectors is
first constructed as an uninterpreted mathematical calculus, the
interpretation can be performed in such a manner that the original
language of physics is extended by the inclusion of the calculus of
vectors. Because the vector symbols are used in the new language in
conjunction with the other linguistic symbols, they have themselves
gained a meaning within the physical language. In the same way,
any system of geometrical axioms can first be given as an isolated
calculus, and the various possible interpretations may be represented
as different translations into the language of physics (see § 7 1 e). If in
this case the terminology of geometry is retained, then it is a ques-
tion of a translation into a congruent sub-language of a new language
constructed from the old language of physics by the inclusion of
geometry.
In order to establish a particular interpretation of the language
Sj, that is to say a particular translation of into Sg, it is not
necessary to give the correlates of all symbols or of all sentences of
Sj. It is sufficient to state the correlates of certain expressions ; in
many cases, for example, it is sufficient to state the correlates of
certain descriptive sentences of a simple form, in which not even
all undefined symbols of Sj need occur. In this way, in connection
with the transformation rules of S^, the whole translation is uni-
vocally determined ; or, more exactly, any two translations which
230
PART IV. GENERAL SYNTAX
have those correlates in common coincide in content. It is cus-
tomary in the construction of a symbolic language, particularly in
logistics, to give an interpretation by means of an expository text,
and hence by means of a translation into the ordinary word-
language. And generally it is also customary to state many more
correlates than are necessary. This is certainly useful for facili-
tating comprehension; and in introducing Language I we have
proceeded in this manner ourselves. But it is important to realize
that interpretative statements of this kind are in most cases over-
determined.
Examples: i. Let the descriptive Language II contain one-termed
predicates ‘ Pi Pk * as the only undefined descriptive symbols.
Then for a complete interpretation of Language II such stipulations
as the following are sufficient; (i) *0* shall designate the initial
position, and an St ‘ 0 *i**‘i * with m dashes, the (m+ i)st position in
such and such a series of positions ; (2) * Pi * shall be equivalent in
meaning to ‘red*, ...‘Pk* to ‘blue*; (3) an atomic descriptive sen-
tence of the form ptj (Stj), where pti is an undefined prt, shall mean
that the position designated by Sti has the property designated by
pti. In the sentences for which the translation is hereby determined,
no defined symbols of any kind whatever occur ; further, no variables
(P» f» 3 > f)» hence no operators and, finally, none of the undefined
logical constants ‘ = *, ‘ 3 ‘ K ‘ ‘ v ‘ ‘ D *. In spite of this,
the interpretation of all the remaining sentences of II is also de-
termined by the above stipulations ; that is to say, for the correlate
of any other sentence of Language II, the only choice is between
equipollent sentences of that sub-language of English into which
Language II is reversibly transformed. Thus ‘Pi(0) vPi(O') *, for
instance, must be translated into : ‘ the first or the second position,
or both, are red* (or into a sentence which is equipollent to this
one). Or again, for example; ‘(a:) (PiC^c))* must be translated into
‘all positions are red*; for it follows from the transformation rules
of Language II that ‘ v * is a symbol of proper disjunction and ‘ (x) *
a proper universal operator.
2. Let III be the logical sub-language of II. Hi is to be inter-
preted by means of a translation in respect of expressions into a
suitable other language ; and, by this translation, a correlate is to be
given for every pr and for every fu. [This requirement is intended to
secure that £^1 is a translation in the ordinary sense ; if the require-
ment is not stated, then the trivial translation may be taken in which
the correlate of every analytic sentence is ‘ 0 = 0 *, and the correlate
of every contradictory sentence ‘ ^(0 = 0)*.] We shall only give the
correlates of two symbols : ‘ 0 ’ will be translated into ‘ 0 * and ‘ • *
into ‘ + 1 *. In this way, the interpretation of the whole of Language
III, which contains classical mathematics, is established.
231
§62. THE INTERPRETATION OF A LANGUAGE
From the standpoint of interpretation, it is characteristic of the
undefined descriptive symbols that their interpretation, even after
that of the other symbols, is still arbitrary within a wide domain
(arbitrary, that is to say, when we merely consider the syntax of
the isolated language the choice can then be limited by further
conditions). Thus, for instance, it is not determined by the trans-
formation rules of Language II and the interpretation of the other
symbols, whether ‘P^’ is to be interpreted, say, by ‘red’ or by
‘green’, or by the designation of any other property of positions.
In most of the symbolic languages even expressions which are
interpreted by their authors as logical belong to the descriptive
expressions as understood in general syntax. The majority of the
usual systems are interpreted by their authors as logical languages ;
but since commonly only d-rules are laid down, these languages
are for the most part indeterminate and therefore descriptive. In
consequence, for certain expressions of these languages, even if the
other expressions are interpreted according to the statements of
the authors, interpretations are possible that are essentially different
from one another.
Example: The universal operator with a numerical variable 3 is a
proper universal operator in Languages I and II, but in the usual
languages — for instance, in [Princ, Math.] — it is an improper one
(see p. 197), because these languages contain only d-rules. Thus, in
the usual languages there are sentences that are indeterminate, and
therefore designated by us as descriptive, although they are inter-
preted by their authors as logical sentences. In order to remain
within the framework of our own symbolism, instead of considering
one of the earlier systems, we will consider that of Language Ild,
which results from II by limitation to the d-rules (but which must
contain all the definitions stated previously in Language I). The sen-
tence (5, which is analytic but irresoluble in II (§ 36), is thus in I Id
an indeterminate sentence. The universal operator (5) in both [Princ.
Math.] and I Id is not logical hut descriptive. By this nothing is said
against the usual translation, in which the correlate of © is a logical
sentence (for example, the identically worded sentence in II), and
the correlate of (3) is a logical expression (for example, a proper univer-
sal operator in II). The fact that (5 and (3) arc descriptive only means
that in addition to this usual translation others are possible, amongst
them some in which the correlates of © and (3) are descriptive. We
will illustrate this for the universal and existential operators by an
example. Let the pri and pr2 be the only undefined descriptive
symbols of I Id and II. We will interpret I Id by means of two dif-
ferent translations into II, Qi and G2- For ^^od Q2» we determine :
232
PART IV. GENERAL SYNTAX
first, that the correlates of all sentential junctions shall be these junc-
tions themselves ; second, that the correlates of all atomic sentences
shall be these sentences themselves. Hence the correlates of all
molecular sentences are also these sentences themselves. We will
now show that G, and Q2 nevertheless still be essentially dif-
ferent from one another — that is to say, that^they need not coincide
in content. Let the Gi-correlate of every sentence be that sentence
itself; this is the ordinary interpretation, in which the improper
universal operator (31) of I Id is interpreted by means of a proper
universal operator of II. Let Sj be (3i) (pTi(3i)); and let G2[Si]
be (3i) (pri(3i)) .prafs). This sentence is (in II) obviously richer in
content than Qi[ 3 i], namely, Sj itself. Let S2 be (3 3i) (P^i( 3 i));
and let be (3 3i) (pr, (31)) v-- ptgCs); tKis sentence is (in II)
obviously poorer in content than Qi[S2], namely, S2 itself. It can
easily be shown that G2 is really a translation (although not an equi-
pollent translation in respect of II), that is, that by means of G2 the
consequence-relation in I Id is transformed into the consequence-
felktion in II. For example, let 83 be pti( 0 ti); then 83 is a conse-
quence of Si, and S2 a consequence of S3; correspondingly,
— i-c* a consequence of the previously given G2[Si];
and the given ^2(02] is a consequence of G2[S3] — i.e. S3. The
reason why, in addition to the ordinary interpretation, the essen-
tially different interpretation Go* which interprets the universal and
existential operators descriptively, is also possible, is that the trans-
formation rules of lid only determine that every sentence of the
form ptj (St) is a consequence of (31) (ptj (3i)), but do not determine
whether this universal sentence is equipollent to (as in the usual
interpretation Gi), or richer in content than (as in the case of Gg),
the class of sentences of the form pri(St).
Other examples of descriptive symbols that are interpreted by
their authors as logical are the intensional sentential junctions treated
of by Lewis and others. (There are, however, also intensional sen-
tential junctions that are logical.)
Let S be a descriptive language for which an interpretation has
been given in the ordinary way in the words of an expository text.
In judging of this interpretation we must, then, distinguish (as the
example just examined shows) between interpretations by means of
descriptive expressions and interpretations by means of logical ones.
(1) Interpretations by means of descriptive expressions generally
yield something new which has not already been given in the con-
struction of the calculus; they are (to a certain extent) necessary
for the establishment of an interpretation of the calculus.
(2) Suppose the expression of the calculus is interpreted by a
logical expression of the word-language. Here, there are two cases
to be distinguished. (*2 a) is a logical expression in the sense
233
§ 62. THE INTERPRETATION OF A LANGUAGE
understood in general syntax ; then that interpretation may already
be implied in the other interpretations, and if so it only serves as
an explanation, which is theoretically unnecessary, but which
facilitates understanding. (2 b) is a descriptive expression in the
sense understood in general syntax (for example: the universal
operator in lid). Then the interpretation of 9 Ii by means of a
certain logical expression can be replaced by the erection of suit-
able c-rules for S, with the aid of which becomes a logical
expression of the kind intended. [Taking our example: let Ild be
expanded by indefinite c-rules to II ; then, in accordance with the
intended interpretation, (3) will become a proper universal
operator. ]
General syntax proceeds according to a formal method, that is
to say, in the investigation of the expressions of a language it
considers only the order and syntactical kind of the symbols of an
expression. We have already seen that this formal method can also
represent concepts which are sometimes regarded as hot formal
and designated as concepts of meaning (or concepts of a logic of
meaning), such as, for instance, consequence-relation, content,
relations of content, and so on. Finally we have established the
fact that even the questions which refer to the interpretation of a
language, and which appear, therefore, to be the very opposite of
formal, can be handled within the domain of formal syntax.
Accordingly, we must acknowledge that all questions of logic
(taking this word in a very wide sense, but excluding all empirical
and therewith all psychological reference) belong to syntax. As
soon as logic is formulated in an exact manner^ it turns out to be
nothing other than the syntax either of a particular language or of
languages in general.
{e) EXTENSIONALITY
§63. Quasi-Syntactical Sentences
We are now going to introduce a number of concepts which are
necessary for the discussion of the problem of extensionality, for
the logic of modalities, and, later on, for the analysis of philo-
sophical sentences. We shall first explain these concepts in an in-
formal and inexact manner. Let B be a domain of certain objects
234
PART IV GENERAL SYNTAX
whose properties are described in the object-language Sj. Assume
that there exists in reference to B an object-property Ei, and in
reference to a syntactical property of expressions Eg, such that
always and only when Ej qualifies an object, Eg qualifies the ex-
pression which designates that object. We^ shall call Eg the syn-
tactical property correlated to E^. Ej is then a property which is,
so to speak, disguised as an object-property, '"it which, according
to its meaning, is of a syntactical character; we therefore call it a
quasi-syntactical property (or sometimes a pseudo-object-pro-
perty). A sentence which ascribes the property E^ to an object c
is called a quasi-syntactical sentence; such a sentence is trans-
latable into the (proper) syntactical sentence which ascribes the
property Eg to a designation of c.
. Examples: i. 'Irreflexiveness. Let St be a descriptive L-Ianguage
(like I and II) with a symbolism similar to that of II, but a name-
language, and let it be concerned with the properties and relations
of the persons living in the district B on a certain day. * Shav (a, b) *
will mean: ‘a shaves b* (on the day in question). Wc define the
2 pr'‘Irr’ as follows: ‘ Irr(F) = (jc) ('---F(Jl[:,;v))^ or, in words: ‘a re-
lation P is called irreflexive when no object has this relation to itself’.
*Irr(Shav)’ is thus equipollent to ‘(jc) ('^Shav(jc, :v))’ (Si). Sj
means that in B, on that particular day, no one shaves himself;
whether that is the case or not cannot be deduced from the trans-
formation rules of Si; Si is synthetic. Let Si contain, in addition,
the 2 pr' ‘LIrr’; ‘Llrr(P)*, or, in words, ‘P is L-irrcflexive (or
logically irreflexive)’, means that P is irreflexive by logical necessity,
that is, ‘LIrr’ must be so defined that ‘Llrr(P)’ is only analytic
when ‘(a*) ( P (.V, a:)^ ’ is analytic, and otherwise it is contradictory.
Then ‘Llrr(Shav)’ is contradictory, since Sj is not analytic but
synthetic. Let ‘ Broth ’ be so defined that ‘ Broth (a, b) * means “ a is
a brother of b”. Then ‘Irr (Broth)’ is analytic, and consequently
‘ LIrr (Broth) ’ is also analytic. ‘Irr’ and * LIrr ’ are predicates of Si.
Let the syntax-language S3 of Sj be a word-language ; we now define
the predicate ‘ L-irreflexive ’ in Sg as follows : a two-termed predicate
pti of Si is called L-irreflexive when (Uj) pri(Di, Dj)) is analytic.
According to this, ‘Shav’ is not L-irreflexivc, but ‘Broth’ is. In a
language which contains both Si and 83 as sub-languages, for any
predicate ‘P’, the sentence ‘Llrr(P)’ is always equipollent to the
syntactical sentence “‘P’ is L-irreflexivc”. ‘L-irreflexive’ is the
syntactical predicate which is correlated to the predicate ‘LIrr’.
‘LIrr’ is a quaM-syntactical predicate of Si; on the other hand,
‘ Irr ’ is not. ‘ LIrr (Broth) * is a quasi-syntactical sentence of Si ; the
correlated syntactical sentence of Sg is ‘“Broth* is L-irreflexive”;
both sentences are analytic. The same is true of * LIrr (Shav) ’
and ‘‘ ‘ Shav ’ is not L-lrreflexive ”. On the other hand, there are no
§ 63 . QUASI -SYNTACTICAL SENTENCES 23 5
syntactical sentences correlated to the synthetic sentences ' Irr (Shav) ’
and * ^ Irr (Shav) * ; therefore, these are not quasi-syntactical sen-
tences. (Concerning * L-irreflexive see § 71 b,)
2. Implication, In the descriptive L-language Si we shall write
‘Imp(A, B)* instead of ‘AdB*. Further, let there be introduced
into Si (by definition or by primitive sentences) a predicate ‘ Limp *
such that, for any closed sentences * A * and ‘ B ‘ Limp (A, B) * is
analytic if, and only if, ‘ Imp (A, B) * is analytic ; otherwise it is contra-
dictory. Let ‘ Ai * and ‘ Bi * be two closed sentences such that ‘ B| ’
is not a consequence of ‘ Ai *. Then ‘ Imp (Ai, Bi) * is not analytic, and
therefore * Limp (Ai, Bi) ’ is contradictory. Now let ‘A2* and ‘Bo*
be two closed sentences of which ‘Bg* is a consequence of ‘Ao’.
Then ‘ Imp (Aj, B2) * is analytic, and consequently ‘ Limp (Ag, B2) ’
is also analytic. Let the syntax-language Sg of Si be the ordinary
word-language. Then, in a language which contains Si and So as
sub-languages, for any two closed sentences, ‘A’ and ‘B’, ‘Limp
(A, B)* is always equipollent to the syntactical sentence “‘B’ is a
consequence of ‘ A* **. ‘ Limp * is thus a quasi-syntactical predicate
of Si to which the syntactical predicate ‘consequence* is correlated.
As opposed to this, ‘Imp* is not quasi-syntactical. To the quasi-
syntactical sentence ‘ Limp (A2, B2) * is correlated the syntactical
sentence “‘Bg* is a consequence of ‘A2*”; likewise, to the quasi-
syntactical sentence ‘ '^LImp(Ai, Bi)* is correlated the syntactical
sentence “‘Bj* is not a consequence of ‘Ai***. On the other hand,
there are no syntactical sentences correlated to the synthetic sen-
tences ‘ Imp (Ai, Bj) * and ‘ ^ Imp (Aj, Bi) ’ ; consequently these sen-
tences are not quasi-syntactical. The relations in this example, to
which we shall return later in a discussion of the logic of modalities,
are completely analogous to those of the first example.
We now pass from the informal and inexact to the syntactical
discussion of these concepts. Let Sj be any language ; and let 83 be
a logical language. Let be a one-one syntactical correlation be-
tween the expressions of Sj and the expressions of a class itg Sg,
and let the expressions of ftg “®tu which are all isogenous with
one another. Then we shall call 83 a syntax-language of 8^ (with
respect to ; and we shall call Qj [?li] the syntactical designation
of (with respect to Qj). The Sg, or ^r, of 83 to which the ex-
pressions of S{2 suitable as arguments we call syntactical 0g,
or (with respect to Q^). If the expressions of itg are numerical
expressions, we call 83 an arithmetized syntax- language. If S3 is a
sub-language of a language S3, we say that S3 contains a syntax of
Si (with respect to Qj).
An Sg", Sgi, of Sj is called a quasi-syntactical ®g when there
exist an 83, a O,, and a logical Sq”, Sga, which fulfil the following
PART IV. GENERAL SYNTAX
236
conditions: Sj is a sub-language of Sg; contains a syntax of
with respect to ; if is any full sentence of Sgi in Sj, such as
which the arguments are not 93, then is
equipollent in S2 to 692 (Cii[^i]>^^i[?t2]» [^n]); let this be
S 2 . Si is then called quasi-syntactical jn respect of 9Ii, . . . 91^ ;
S2 is called a syntactical sentence correlated to Si (with respect to
Qi) ; 002 Is called a syntactical Sg correlated to Sgi (with respect
to Oj). These definitions also hold for place of
® 9 i. ® 92 -
Let Sg2 be a syntactical Sg which is correlated to Sgj with
respect to jQj. Let SfUi have the form 0 gi( 9 li, ...9I7,), where at
least one of the arguments is a 93; let Sfu2 have the form
Sg2(3ri, ... 91 n)» where 9 I\ (1 = i to w) is Qj [ 91 J if 91 * is not a 93 ;
if 91 , is a 93 „ let 9 r, be a 93 of S2, to the substitution-values of
which belong the Qj-correlates of the substitution-values of 93,.
Then we call Sf U2 a syntactical Sfu correlated to (with respect
to Qj). Let SfU2 be a syntactical Sfu correlated to SfUi- Let
be constructed from 0fu by means of operators, and similarly 02
from 0fu2 by means of corresponding operators. Then we say that
02 is a syntactical sentence correlated to 0 i. Every sentence that
contains a quasi-syntactical 0g, or 0fu, is called a quasi-
syntactical sentence. For compound quasi-syntactical sentences,
the correlated syntactical sentences are constructed in a manner
analogous to the simple cases here described.
Example: Let ‘Pi'(F)’ and be quasi-syntactical 0 fu in
Si* Let the correlated syntactical Sfu be ‘QiW* or ‘Q2(x,3;)*,
respectively. Then the syntactical sentence ‘ (jc) [Qj (jc) D (3 y)
(Q2(x,y))]* is correlated to the quasi-syntactical sentence
‘(F)[Pi(F)D(3m)(P2(F, «))]’.
The difference between the quasi-syntactical sentences and the
others is connected with the difference between syntactical concepts
and the concept ‘ true ’. If one were to take ‘ true ’ as a syntactical
term, then every sentence whatsoever, 0i, in relation to every partial
expression, 9 li, would be quasi-s^mtactical. For 0 ^ is always equi-
pollent to the sentence ‘ 9 li is such that Sj is true.* If Si is a logical
language^ then, with respect to Si, ‘ true * coincides with ‘ analytic *
(that is to say, there are here no synthetic sentences ; see Theorem
52.3). Consequently, in this case, the concept ‘quasi-syntactical*
becomes trivial. For instance, let Si be the logical sub-language of I.
And let pti be ‘Prim*. Then, for every 3 i» the sentence pri( 3 i) of
Sj is equipollent to the sentence of the syntax-language ‘ 3i is such
that pti (3i) is analytic* ; for either both sentences are analytic or both
237
§63. QUASI-SYNTACTICAL SENTENCES
are contradictory. Therefore, ptiOi) is a quasi-syntactical sentence
in respect of 3 i« But iu relation to the descriptive Language I this
is not the case. If fUi is an undefined fub, then (fUj (nu)) is syn-
thetic, and therefore not equipollent to the syntactical sentence
‘fUi(nu) is such that pti (fUi(nu)) is analytic*, for the latter is con-
tradictory. When, in what follows, we establish the fact that certain
sentences in certain languages are quasi-syntactical sentences, this
means that they are still quasi-syntactical even if we expand the
language so that it becomes descriptive (and in such a way that
descriptive arguments for the positions in question exist). [Later,
for instance, we shall assert that the of the logic of modalities arc
quasi-syntactical ; by which we mean also to maintain that they are
still quasi-syntactical even if we extend the calculus of the logic of
modalities by admitting synthetic sentences, also, as arguments. For
the consequence-predicate of the logic of modalities (e.g. for the
symbol of strict implication and similar ones) this is shown by the
example ‘Limp* on p. 235.]
§ 64. The Two Interpretations of Quasi-
Syntactical Sentences
Let the sentence Si of the form SQi (?Ii) be quasi-syntactical, and
let the sentence S2 of the form Sga (^Is) Be a correlated, and hence an
equipollent, syntactical sentence. We will distinguish two possible
interpretations which might here be intended. (This is only a
material, non -formal investigation which serves as a preliminary
to the formal definitions.) In both, is interpreted as a syntactical
designation of the expression ^1, and SQs as a designation of a
syntactical property of expressions. The two cases to be dis-
tinguished are as follows; (i) where Sgi is taken as equivalent in
meaning to Sg2; and (2) where it is not. In the case of the first
interpretation, Sgi as well as Sg2 designates a syntactical property ;
since Si and Sj are equipollent, the equivalence in meaning of the
arguments follows from the equivalence in meaning of the two Sg.
Thus, here 9 Ii, like 2l2> be interpreted as a syntactical designa-
tion of ?li ; designates itself, and is therefore autonymous. [The
term ‘autonymous* has already been explained on p. 156; its
strictly formal definition will be given later.] In the case of the
second interpretation, Sgi designates not a syntactical property
but an object-property, which is attributed to the object designated
by ?li (not to the expression ?li) in the sentence Si- We will in
general assign to the material mode of speech any sentence which
238 PART IV. GENERAL SYNTAX
(like Qi in the second interpretation) is to be interpreted as at-
tributing to an object a particular property, this property being
quasi-syntactical, so that the sentence can be translated into
another sentence which attributes a correlated syntactical property
to a designation of the object in question.^ In contrast with the
material mode of speech of the quasi-syntactical sentences of the
second interpretation we have the formal mode of speech of the
syntactical sentences.
Example: 1 . Quasi-syntactical sentences : {a) autonymous mode of
speech, “Five is a number-word**; (6) material mode of speech,
“Five is a number.** 2. Correlated syntactical sentence: “‘Five* is
a number-word.** (For the sake of simplicity, in i a and 2 we have
taken as pr that are equivalent in meaning the same word, ‘ number-
word *.)
Our task now is to represent formally the difference between
the two interpretations that has just been indicated materially.
Which formal syntactical properties of and ©gg correspond to
the fact that Sg^ is intended as equivalent in meaning to Sgg and
thus as a designation of a syntactical property } It is not necessary
for Sg^ and 3g2 to be synonymous (or L-synonymous) ; for it may
well be that, in spite of their equivalence in meaning, we intend to
admit only Gg^ with autonymous arguments, and not Sg2- In this
case Sgi(?l2) would certainly be equipollent to Ggi('2li); but
Sg2(?li) would not — for it need not be a sentence. But if Sgi is
intended to designate a syntactical properly, and, further, the same
syntactical property as ©Qa* ^^en ©g^ (^Ig) is equipollent to ©ga (2I2).
On the basis of this preliminary consideration, we formulate the
following formal syntactical definitions (for the sake of simplicity
we do so in relation to ©g^ ; the definitions for the case of two or
more argurrients are analogous, likewise those referring to ^r).
Let the sentence ©^ of Sj have the form ©g^ ( 9 Ii) and be quasi-
syntactical in relation to ; and let not be a 93 . Let S2 contain
both Sj and a syntax of Sj with respect to Let ©gg (Qi [?li])
be a syntactical sentence of S2 correlated to ©^ with respect to Qi.
Two cases are to be distinguished: (i) Sgi(Oi[ 9 IJ) is a sentence
of Sg and, in Sg, is equipollent to ©gg (Qi [ 9 Ii]); likewise, for every
5(2 which is isogenous with 5 Ii, Sgi(Qi[ 9 l 2 ]) is equipollent to
©gg (Qi [^2])* t^is case we call 9 Ii autonymous in ©1 (with
respect to and a sentence of the autonymous mode of
§ 64. INTERPRETATIONS OF QUASI -SYNTACTICAL SENTENCES 239
speech (with respect to Qj). (2) The given condition is not
fulfilled. In this case we say that belongs to the material mode
of speech (with respect to Qj). Let Qg be a translation of Si into
S2 in respect of sentences; further, let the Qg'Correlate of every
quasi-syntactical (with respect to Qi) sentence of Sj be a syntactical
sentence correlated to it with respect to Qi ; and let the Qg'Correlate
of every other sentence be the sentence itself. The Qg’^r^^slation
of the sentences of the material mode of speech into correlated
syntactical sentences is called a translation from the material into
the formal mode of speech.
It is to be noted that the differentiation between autonymous
and material modes of speech is concerned with interpretation.
This means that this differentiation cannot be made in relation to a
language Si which is given as an isolated calculus without any in-
terpretation. But it does not mean that the distinction lies outside
the domain of the formal, in other words, of syntax. For, even the
interpretation of a language can be formally represented and thus
be incorporated in the syntax. As we have seen, the interpretation
of a language Si in relation to an assumed language Sg can be
formally represented either by the translation of Sj into Sg, or by
the incorporation of Si as a sub-language in a third language S3,
which is constructed from the language Sg by extension. If Si is a
quasi-syntactical sentence of Si, and if the interpretation of Sj is
formally determined by the fact that Si is a sub-language of a
language Sg which contains also the syntax of Si, then, according
to the definitions just given, it can be determined whether Si be-
longs to the autonymous or to the material mode of speech. But in
practice we are frequently not in a position to make this distinction
with accuracy ; namely, where it is a question of a system Si which
another author has constructed without giving either the trans-
lation of Si into, or its incorporation in, another language also con-
taining the syntax of Si. If in such a case no interpretation what-
soever is given, then the distinction disappears. In the majority of
calculi which have been constructed up to the present, although an
interpretation has been given, it has usually not been done by
means of strict syntactical rules (either incorporating Si in, or trans-
lating it into, some other formally established language Sg), but
only by material explanations, that is to say, by the translation of
sentences of Si into more or less vague sentences of a word-
240
PART IV. GENERAL SYNTAX
language. If we undertake on the basis of such explanations a
translation of into a formally established language S2, we can, at
most, suppose that what was meant by the author has been more or
less accurately expressed, that is, that we have proposed a trans-
lation which deviates less or more from that which the author him-
self would have proposed as a translation of into Sg. When in
what follows we attribute certain sentences of the calculi of other
authors, or of the word-language, either to the autonymous or to
the material mode of speech, it must be noted that this is not in-
tended as an exact and final classification; in the case of the sen-
tences of other calculi, the differentiation depends upon the in-
terpretative explanations given by their authors, and in the case of
the sentences of the word-language upon consideration of the
ordinary use of language. On the other hand, the decision that
certain sentences are quasi-syntactical (not genuinely syntactical)
can be made with the same degree of exactitude with which the
language in question is itself constructed ; in this we need take no
heed of interpretation, whether given materially or formally.
§ 65. Extension AL iTY in Relation to
Partial Sentences
By way of preparation for the definition of extensionality, we will
first examine the definition that has been usual hitherto. An Sfu}
with one variable fj is commonly called extensional (or a truth-
function) in relation to f^, if for any Si and S2 whatsoever, having
the same truth-value, j same
truth-value. If, for instance (in a symbolism like that of II),
*T(j>y is ail Sfu of the kind in question, then ‘T(/))* is called
extensional, if ‘(p=^?) 3 (T(p) = T(^))' (Si) is true. We must
formulate this definition differently ; we do not use the term ‘ true *
because it is not a genuine syntactical term ; further, we will not make
the limiting assumption that sentential variables and symbols of
proper equivalence and implication exist in S. Since Si must be
not only true (indeterminately) but valid, we can replace the given
condition by the following: for any closed sentences whatsoever,
say ‘A’ and ‘B*, ‘T(A)=T(B)* (Sg) must be a consequence of
' A = B ’ (Sg). The implication having been eliminated, we will now
§65. EXTENSIONALITY IN RELATION TO PARTIAL SENTENCES 24 1
eliminate the equivalence also. S2 property that * B ’ is a
consequence of S2 * A and ‘ A’ a consequence of Sg and ‘ B
Further, Sg is the poorest in content of the sentences having this
property, that is to say, if any likewise possesses the property in
question, then Sg is a^ consequence of fti; hence, S3, if a conse-
quence of Sg, is also a consequence of These considerations
lead us to the statement of the following definitions.
In analogy with the previously defined concepts — absolute
concepts, as it were — of the equipollence of two R (or two S), the
coextensiveness of two Sg (or Sfu or the synonymity of two
'21, and the identity of the course of values of two '2lg (or '2lfu
or gu), we will now define the corresponding relative terms in
relation to a sentential class. Si and Sg are called equipollent (to
one another) in relation to R^ if Sg is a consequence of + {( 5 i]
and Si a consequence of i^i + ISg}. SQi and Sgg are called
coextensive (with one another) in relation to 5^i if every two full
sentences with equal arguments are equipollent in relation to ;
similarly for two Sfu or two (isogenous) Two isogenous ex-
pressions 9 Ii and are called synonymous in relation to i^i, if every
Si is equipollent to relation to R^. We
say that '2lgi and ^Igg have the same course of values in relation to
when every two full expressions with equal arguments are synony-
mous in relation to R ^ ; likewise for two '2lfu or gu.
Theorem 65.1. (a) If two S are equipollent, then they are also
equipollent in relation to every R. {b) Analogously for co-
extensiveness. (c) Analogously for synonymity, {d) Analogousl}
for identity of the course of values.
Theorem 65.2. (a) If Si and Sg are equipollent in relation to a
valid Ri then they are equipollent (absolutely). (6) Analogously
for coextensiveness, (r) Analogously for synonymity, (d) Ana-
logously for identity of the course of values.
Extensionality in relation to partial sentences. Si is called ex-
tensional in relation to the partial sentence Sg if for any closed S3
and any fti such that Sg and S3 are equipollent in relation to
Si and Si ra are always equipollent in relation to Ri, An Sgi
to which S are suitable as arguments is called extensional if, for any
closed Si and Sg and any Ri such that Si and Sg are equipollent
PART IV. GENERAL SYNTAX
24Z
in relation to Sfli (0i) and Sgi (S2) are always equipollent in
relation to Correspondingly in the case of an Sfu or to
which S are suitable as arguments. If every sentence of S is ex-
tensional in relation to every closed partial sentence, then we call
S extensional in relation to partial sentences. 'Intensional’ is to
mean the same as ‘ not extensional * (in the different connections).
[‘ Intensional * as we use it means nothing more than this, and in
particular it means nothing like * related to sense etc. ; in many
authors the word has a meaning of this kind, or even a mixture of
the two meanings (see § 71).]
Theorem 65.3. If S is extensional in relation to partial sen-
tences, then all Sg, Sfu, and of S to which S are suitable as
arguments are extensional.
. Theorem 6s»4* Let S be extensional in respect of partial sen-
tences. (a) If two closed S are equipollent in relation to 5 ^^, then
they are also synonymous in relation to (b) If two closed S are
equipollent, then they are also synonymous, (c) If two closed
whose arguments are S are coextensive in relation to 5li, then they
are also synonymous in relation to Siy (d) If two closed "ipr whose
arguments are 0 are coextensive, then they are also synonymous,
(e) If two closed Ju whose arguments are S have the same course
of values in relation to ftj, then they are also synonymous in relation
to 5li. (/) If two closed 5u whose arguments are 0 have the
same course of values, then they are also synonymous.
Hieorem 65.5^ Sentential junctions. If a 93 f or a n! possesses a
characteristic, then it is extensional ; and conversely. — Thus, proper
negation, proper implication, etc., are extensional.
Theorem 65.6. If S is extensional in respect of partial sen-
tences, then all are extensional.
Theorem 65.7. Let 35 fi be a proper equivalence in S. Then it is
true that: {a) 0^ and 02 are always equipollent in relation to
(^) S extensional in relation to partial sentences
if, and only if, for any closed 0i, 02, and 03, 95 fi ^03, 03 j
is always a consequence of 33 li( 0 i, 02). (c) Further, let SSIg be
a proper implication in S; then S is extensional in relation to
partial sentences if, and only if, for any closed 0^, 02, and 03,
§65. EXTENSIONALITY IN RELATION TO PARTIAL SENTENCES 243
Theorem 65.8. Let ‘ ’ be a symbol of proper equivalence in
S. (a) If, for closed S2 @2 ~ ^ ^ consequence of Silt
but Si = Si ra is not a consequence of then is intensional
in relation to Sg. (b^ If, for two closed sentences Sg and S3,
02 = 08 is valid but valid, then Sj is in-
tensional in relation to Sg.
SgJ is called an Sg of identity^ if every two possible closed argu-
ments and ^Ig are always synonymous in relation to Sg^ (^Ij, ^Ig).
An 0 g of identity, Sg^, is called an 0g of proper identity if for
every two possible closed arguments ?lg, which are synonymous
in relation to 0gi(5Ii,2l2) is always a consequence of Ai;
otherwise it is called an Sg of improper identity. If Sg^ is an 0 g
either of proper or of improper identity, then Sg^ (9li, Slg) is called
a sentence of proper (or improper, respectively) identity (or an
equation) for and ^Ig. A prf is called a symbol of •proper or
improper identity (or predicate of identity, or symbol of equality)
in general, or for all expressions of the class Ai, if the sentence
pri(?li,?tg) is a sentence of proper (or improper, respectively)
identity for 3Ii and ^Ig for any closed or any closed 91 of Ai
respectively. (S may, for instance, contain different symbols of
identity for 3» ®i ®i^d ^r.)
Theorem 65.9- Let 0 i be a sentence of identity for the closed
expressions and ^Ig. (a) 9 li and ^Ig are synonymous in relation to
01. (b) If 01 is valid, then 9 Ii and ^tg are synonymous (absolutely).
Theorem 65.10. Let S be extensional in relation to partial
sentences, {a) If 95 fi is a proper equivalence, then for any two
closed sentences 0i and ©g, 2 )fi( 0 i, Sg) is always a sentence of
proper identity for 0 i and 02- {b) A symbol of proper equi-
valence is a symbol of proper identity for sentences.
§ 66. Extensionality in Relation to
Partial Expressions
Here we shall again start from the usual definition (using the
S5rmbolism of II). It is customary to call an 0 fuj with a variable
pi, say ‘M(F)*, extensional in relation to ‘P* if
‘ (*) {F(x) = G {x))o (M (F) = M (G))'
16-2
244
PART IV. GENERAL SYNTAX
is true. We can, as previously, alter the formulation of the con-
dition thus: for any ‘Pi* and ‘Pg*, ‘ M (Pj) = M (Pj) * must always
be a consequence of ‘(*) (Pi(^) = P2W)*- With this as a basis,
we now give the following definitions.
Extensumaliiy in relation to partial expressions. Let occur in
Si*, Si is called extensional in relation to ^ti if for any closed
and any Ri such that ^ti and ^12 are coextensive in relation
to ill, ®i ®i
are always equipollent in relation to Mi.
Let 3 fUi occur in Si ; Si is called extensional in relation to gUi if,
for any closed 5U2 and any Mi such that gui and 5^2 have the
same course of values in relation to Mi, Si and Si
are equi-
pollent in relation to Mi. If Si is extensional in relation to all the
closed S, ^r, and gu which occur in Si, Si is called extensional.
An Sgi, to ^hich ^r, Sfu, or S are suitable as arguments, is called
extensional if every full sentence of SQi with qlosed arguments is
extensional in relation to every argument. Correspondingly for
every SfUi or ^ti to which ^r, 5 u, or S are suitable as arguments.
If every sentence of S is extensional in relation to every closed
partial expression (or ^u) then S is called extensional in relation
to (or respectively). If S is extensional in relation to partial
sentences, to ^r, and to ^u, then S is called extensional.
Theorem 66.i. {a)lfS is extensional in relation to ^r, then two
closed which ^re coextensive (absolutely or in relation to Mi)
are always (absolutely or in relation to Mi, respectively) synony-
mous. (Jb) If S is extensional in relation to JJfu, then two closed
which have the same course of values (absolutely or in relation to
Ml) are always (absolutely or in relation to Mi, respectively)
synonymous.
Examples: The languages of Russell and of Hilbert and our own
Languages I and II are extensional in relation to partial sentences.
That is shown, for instance, by the criterion of Theorem 65.7 c (cf.
Hilbert [Logi^], p. 61). The symbols of equivalence in these lan-
guages are symbols of proper equivalence and hence, according to
Theorem 65.106, they are also symbols of proper identity for S.
The form of the language will be simpler if only one symbol of
identity is used (as in I and II, and in contrast with Russell and
Hilbert), the same for S as for 3 > and so on. If from Russell’s
language R we construct a new language R^, by extending the rules
of formation to adinit of undefined pib with 6 as arguments, then
§66. EXTENSIONALITY IN RELATION TO PARTIAL EXPRESSIONS 245
is no longer necessarily extensional in relation to partial sen-
tences; in order to guarantee extensionality here also, we can pro-
ceed, for example, by admitting 0 =6 as a sentence, and (in analogy
with PSII 22, see below) stating a new primitive sentence as follows :
* (p = q) 0 (p = q)*- If the extended language IT is constructed from
II in the same way, then it is extensional in relation to partial sen-
tences. Here no new primitive sentence is necessary, since we use
the symbol of identity as symbol of equivalence, so that the above
sentence of implication is demonstrable.
Languages I and II are also extensional in general. In II the ex-
tensionality in relation to and is guaranteed by PSII 22 and
23 (see p. 92). In the case of the other languages, the question of
extensionality in relation to and can only be decided after
further stipulations have been made, especially regarding what
undefined "ptb (for «> i) are to be admitted.
The languages of Lewis, Becker, Chwistek, and Hey ting are
intensionaly for partial sentences as well as for the rest (see § 67), •
§ 67. The Thesis of Extensionality
Wittgenstein {\Tractatus\y pp. 102, 142, 152) put forward the
thesis that every sentence is “a truth-function of the elementary
sentences** and therefore (in our terminology) extensional in re-
lation to partial sentences. Following Wittgenstein, Russell
{\Introduction\y pp. 13 ff. ; [Princ. Math!\ Vol. i, 2nd edition, pp.
xiv and 659 ff.) adopted the same view with regard to partial
sentences and predicates ; as I also did, but from rather a different
standpoint {[Aufbau\y pp. 59 ff.). In so doing, however, we all
overlooked the fact that there is a multiplicity of possible languages.
Wittgenstein, especially, speaks continually of “the** language.
From the point of view of general syntax, it is evident that the
thesis is incomplete, and must be completed by stating the lan-
guages to which it relates. In any case it does not hold for all
languages, as the well-known examples of intensional languages
show. The reasons given by Wittgenstein, Russell, and myself, in
the passages cited, argue not for the necessity but merely for the
possibility of an extensional language. For this reason we will now
formulate the thesis of extensionality in a way which is at the same
time more complete and less ambitious, namely: a universal
language of science may be extensional \ or, more exactly: for every
given intensional language Sj, an extensional language S2 may be
constructed such that can be translated into S2. In what follows
246 PART IV. GENERAL SYNTAX
we shall discuss the most important examples of intensional sen-
tences and demonstrate the possibility of their translation into
extensional sentences.
Let us enumerate some of the most important examples of in-
tensional sentences, ‘ A * and * B * are abbreviations (not designations)
for sentences, e.g. “It is raining now in Paris**, etc. i. Russell
([Princ, Math.]t Vol. i, p. 73 and [Math, Phil,]^ pp. 187 ff., and
similarly Behmann [Logik], p. 29) gives examples of approximately
the following kind: “Charles says A**, “Charles believes A**, “it is
strange that A **, “ A is concerned with Paris ’*. Incidentally Russell
himself later, influenced by Wittgenstein *s opinions, rejected these
examples, and asserted that their intensionality was only ap-
parent ([Princ. Math,], Vol. i, 2nd edition. Appendix C). We
prefer to say instead that these sentences are genuinely intensional
but are translatable into extensional ones. 2. Intensional sentences
concerning being-contained-in and substitution in relation to ex-
pressions : “ (The expression) Prim (3) contains (the expression) 3 ** ;
“ Prim (3) results from Prim fx) by substituting 3 for x”. Sentences
of this kind (but written in symbols) occur in the languages of
Chwistek and Hey ting, 3. Intensional sentences of the logic of
modalities: “A is possible**; “A is impossible**; “A is necessary**;
“ B is a consequence of A ** ; “ A and B are incompatible **. Sentences
of this kind (in symbols) occur in the systems of the logic of modali-
ties constructed by Lewis, Becker, and others. 4. The following
intensional sentences are akin to those of the logic of modalities:
“Because A, therefore B**; “Although A, nevertheless B**; and the
like. That any sentence Si of the examples given is intensional in
relation to ‘ A * and ‘ B * follows easily from the criterion of Theorem
65.8a. If, for instance, ‘A* is analytic and ‘C* is synthetic, then
‘ A = C * is a consequence of ‘ C ’ ; but the false sentence “ A is neces-
sary = C is necessary** is not a consequence of ‘ C*. These examples
will be discussed in greater detail in what follows.
The above examples appear at first glance to be very different
in kind. But, as a closer examination will show, they agree with
one another in one particular feature, and this feature is the reason
for their intensionality: all these sentences are quasi-syntactical sen-
tences and, in particular, they are quasi-syntactical with respect to
those expressions in relation to which they are intensional. With
the establishment of this characteristic, the possibility of their trans-
lation into an extensional language is at once given, inasmuch,
namely, as every quasi-syntactical sentence is translatable into a
correlative syntactical sentence. That the syntax of any language
(even an intensiqnal one) can be formulated in an extensional
language is easy to see. For arithmetic can be formulated to any
§67. THE THESIS OF EXTENSIONALITY 247
desired extent in an extcnsional language, and hence an irith-
metized syntax also. Incidentally this is equally tme of a syntax in
axiomatic form.
What we have said holds for all examples of intensional sen-
tences so far known. Since we are ignorant of whether there exist
intensional sentences of quite another kind than those known, we
are also ignorant of whether the methods described, or others, are
applicable to the translation of all possible intensional sentences.
For this reason the thesis of extensionality (although it seems to me
to be a fairly plausible one) is presented here only as a supposition,
§ 68. Intensional Sentences of the Autonymous
Mode of Speech
Some of the known examples of intensional sentences belong to
the autonymous mode of speech. When translated into an ex-
tensional language, they are transformed into the corrdated syn-
tactical sentences. We will first of all examine the converse process,
namely, the construction from an extensional syntactical sentence of
an intensional sentence with an autonymous expression. By this
means the nature of these intensional sentences will become clear.
Let Si and Sg be extensional languages; and let Sg contain Sj
as a sub-language and the syntax of by virtue of Qj. Let be
an S, or gu of S^, and S2 (in Sg) have the form ‘iPr2(Qi n®i])-
In material interpretation : [?IJ is a syntactical designation of
^1; S2 ascribes to 9ti a certain syntactical property expressed by
^r2. in general not a sentence of S2. Now, out of S2,
we construct an extended language S3 (that is to say, Sg is a proper
sub-language of S3). The rules of formation are extended as
follows: in S3, for every 9I3 which is isogenous with ?[i in S^,
^^^2(^3) is ^ sentence, and hence ^r2(?Ii) also (let this be Sj);
further, the rules of transformation are extended as follows : in S3,
for every which is isogenous with in Sj, ^r2(^3) is equi-
pollent to *?5r2 (QiPy), and therefore Si is also equipollent to
’iPrg (Qi py ) (this is Sg)- Then, according to the criterion given
earlier (p. 238), 2 fi is autonymous in Si- A sentence which is
formulated like Si is in general intensional in respect of Slj.
Example: Let Si be I. As syntax-language in Sg we will take the
word-language. Let the Qi-correlates (the syntactical designations)
PART IV. GENERAL SYNTAX
248
be formed with inverted commas. T^et be ‘0ll = 2*, and ac-
cordingly “ O' • = 2 * *. Let S2 be “ 0" = 2 * is an equation *. Then
0i is ‘ 0" = 2 is an equation *. For S3 we stipulate that Si and S2 be
mutual consequences of one another; and likewise, corresponding
other sentences with the same ^r. Then ‘0" = 2* is autonymous in
0,, and, according to Theorem 6s.8b, Sj is iqtensional in relation to
‘0" = 2\ For let ^3 be ‘Prim (3)*; then ^ = % is analytic but
‘ Prim (3) is an equation ’ (84), because it is equipollent to ‘ ‘ Prim (3) *
is an equation*, is contradictory; hence, since Si is analytic, 0i= 84
IS contradictory.
Now some of the examples of intensional sentences previously
mentioned have the same character as the intensional sentences
constructed in the way here described : their intensionality is due
to the occurrence of an autonymous expression. We will cite some
examples of this, at the same time giving the correlated syntactical
sentences. The latter may belong to an extensional language.
[Sentences i b and 2 b belong to descriptive syntax, 3 46, and
5 A to pure syntax. The preceding investigations and definitions
have all been given in relation to pure syntax only; they may,
however, be correspondingly extended to apply to descriptive
syntax.] To interpret these sentences as belonging to the autony-
mous mode of speech seems to me to be the natural thing, espe-
cially in the case of 4 and 5 a. However, if anyone prefers not
to ascribe one of them (say 2 a or 3 a) to the autonymous mode of
speech, he is at liberty to do so ; the sentence in question will then
belong to the material mode of speech. The only essential points
are: (i) these intensional sentences are quasi-syntactical ; and
(2) they can (together with all other sentences of the same lan-
guage) be translated into extensional sentences, namely, into the
correlated syntactical sentences.
Intensional sentences Extensional sentences
of the autonymous mode of syntax
of speech
Let ‘A * be an abbreviation (not a designation) of some sentence.
1 a. Charles says (writes, reads) i b. Charles says ‘A’.
A.
2 a. Charles thinks (asserts, be- 2 b, Charles thinks ‘ A *.
lieves, wonders about) A.
[Of the same kind is the following: “it is astounding that. . . ’*, that
is to say: “many wonder about the fact that. . . **.]
§68. INTENSIONAL SENTENCES OF THE AUTONYMOUS MODE 249
3 a. A has to do with Paris. 3 b. * Paris * occurs in a sentence
which results from *A* by
the elimination of defined
symbols.
4a. Prim (3) contains 3. 4^- ‘3’ occurs in ‘Prim (3)’.
5 a. Prim (3) results frpm 5 6. ‘ Prim (3) * results from
Prim (x) by the substitution ‘ Prim (x) * by the substitu-
of 3 for X. tion of ‘ 3 * for ‘ x *.
We have here interpreted the previously mentioned (p. 246)
examples of intensional sentences put forward by Russell, Chwistek,
and Hey ting, as sentences of the autonvmous mode of speech. This in-
terpretation is suggested by the relevant indications given by the
authors themselves. Russell’s sentences are already presented in the
word-language; and for the sentences of Chwistek and Heyting,
which are formulated in symbols, the authors themselves give para-
phrases in the word-language corresponding to 4a and 5 a. ^ .
Chwistek’s system of so-called semantics is, on the whole, dedi-
cated to the same task as our syntax. But Chwistek throughout em-
ploys the autonymous mode of speech (apparently without being
aware of it himself). He uses as the designation of an expression
with which a sentence of semantics is concerned either this ex-
pression itself or, alternatively, a symbol which is synonymous with
it (and is thus, originally, not a designation but an abbreviation for
it). As a result of the employment of the autonymous mode of
speech, many sentences of Chwistek’s semantics are intensional.
Because of this, he has come to the conclusion that every formal
(Chwistek says “nominalistic”) theory of linguistic expressions
must make use of intensional sentences. This view is refuted by the
counter-example of our syntax, which, although strictly formal, is
consistently extensional (this is most clearly seen in the formalized
syntax of I in I, in Part II). The fact that Chwistek believed himself
forced to abandon the simple rule of types for his semantics and to
return to the branched rule (see § 60 a), was also, in my opinion,
only a consequence of his use of the autonymous mode of speech.
Heyting gives as the word-translation of certain symbolic ex-
pressions of his language: “the expression which results from a
when the variable x is replaced wherever it appears by the com-
bination of symbols p ({Math, i], p. 4) and: “g does not contain x”
{[Math, i], p. 7). Such formulations, like our examples 40 and 5 a,
belong, without any doubt, to the autonymous mode of speech.
But even the sentential calculus of Heyting’s system [LogiK\ contains
intensional sentences ; sentential junctions which can be shown to
possess no characteristic are used (see p. 203). These circumstances
make it natural to suppose not only that the whole system can be
translated by us into a system of syntactical sentences, but also that
this was in a certain sense the author’s intention. “ In a certain
sense” only, because the distinction between the object- and the
2S0
PART IV. GENERAL SYNTAX
syntax-languages is nowhere explicitly made ; so that it is not even
clear which language it is whose syntax is supposed to be represented
in the system. According to [Grundlegung], p. 113, the assertion
of a sentence (which is formulated symbolically by placing the
symbol of assertion in front of the sentence) is “the establishment
of an empirical fact, namely the fulfilment of^the intention expressed
by the sentence** or of the expectation of a possible experience.
Such an assertion may mean, for example, the historical circum-
stance that I have a proof of the proposition in question lying in
front of me. According to this, the assertions in Heyting’s system
should be interpreted as sentences of descriptive syntax. On the
other hand, Godel [Kolloquium 4], p. 39, gives an interpretation of
Heyting*8 system in which the sentences of the system would be
purely syntactical sentences about demonstrability ; “A* is de-
monstrable ’ is formulated by means of ‘ BA *, and consequently in
the autonymous mode of speech.
§ 69. Intensional Sentences of the Logic
OF Modalities
We shall now give some further examples of intensional sen-
tences together with their translation into extensional syntactical
sentences. By means of this translation the intefisional sentences are
shown to be quasi-syntactical. Sentences i a to 4a contain terms
that are usually known as modalities [‘ possible \ ‘impossible*,
‘necessary*, ‘contingent* (in the sense of ‘neither necessary nor
impossible*)]. Sentences 5a to 7^ contain terms that are similar
in character to these modalities, and are therefore treated by the
newer systems of the logic of modalities (Lewis, Lukasiewicz,
Becker, and others) together with them. In these systems, the
modal sentences are symbolically formulated in approximately the
same way as our examples 16 to 7^. Examples are intensional
sentences of the ordinary word-language which we add here be-
cause, as the syntactical translation shows, they are akin to the
modal sentences. ‘ A * and ‘ B * arc here sentences — i.e. abbreviations
(not designations) of certain sentences (such as synthetic sentences)
either of the word-language or of a symbolic language.
Intensional sentences of the
logic of modalities
1 a, A is pos-
sible.
2 0. A« A is
impossible.
lb. P(A).
ib. I(A.-wA);
'-P(A.-'A).
Extensional sentences of
syntax
ic. ‘ A * is not contradictory.
zc. ‘A * — ^ A * is contradictory
§69. INTENSIONAL SENTENCES— LOGIC OF MODALITIES 25 1
3 a. A V ^ — A is
necessary.
4. a. A is co7i~
tingent.
5 a. A strictly
implies B ; B
is a conse-
quence of A.
6 a. A and B
are strictly
equivalent.
7 a. A and B
are compat-
ible.
8a. Because A, therefore B; A, 8c. ‘A* is analytic, ‘B* is an
hence B. L-consequence of ‘ A *, * * is
analytic. (‘ A * is valid, ‘ B Ms
a consequence of ‘ B * is
valid.)
Since the terms used in the logic of modalities are somewhat vague
and ambiguous, it is also possible to choose other syntactical terms
for the translations ; in 2 c, for instance, instead of ‘ contradictory * we
may put ‘ contravalid ‘ L-refutable *, or ‘ refutable *. Similarly in the
other cases, instead of the L-c-term we can take the general c-term,
the L-d-term, or the d-term. With regard to 8 c, in the majority of
cases the general c-term (or the P-term) is perhaps more natural as
an interpretation of 8 a than the L-term. The difference between the
so-called logical and the so-called real modalities can be represented
in the translation by the difference between L- and general c-terms
(or even P-terms) :
ga. A is logically impossible. gc. ‘ A ’ is contradictory.
10 a. A is really impossible. 10 Ci. ‘ A * is contravalid.
10 C2. ‘AMs P-contravalid.
The translation of 10 a depends upon the meaning of ‘really im-
possible ’. If this term is so meant that it is also to be applied to cases
of logical impossibility, then the translation loci must be chosen;
otherwise locg. Analogous translations may be given for the three
other modalities — for ‘ logically (or “ really”, respectively) possible
‘necessary*, and ‘contingent*.
That sentences la to 10 a and 16 to 76 are intensional is easily
seen. [Example: Let ‘Q* be an undefined pr^, and ‘ = * a symbol
of proper equivalence. Let ©j be ‘Prim(3) = Q(2)*; ©2 be:
‘ Prim (3) is necessary * ; and ©3 : ‘ Q (2) is necessary *. Then ©2 = ©3
cannot be a consequence of ©1 (for ©1 is synthetic, ©2 analytic, and
©3 contradictory, and hence ©2 =©3 is contradictory). Therefore
(by Theprem 65.76) ©2 is intensional in relation to ‘Prim (3)*.]
36. N(Av^A); 3 c. ‘ A V A* is analytic.
-'P'-(Av-^A).
46. ^ N (A) . 4 c. ‘ A * is svnthetic. (‘ A * is
1 (A); neither analytic nor contra-
P (A) • P ( ~ A). dictory ; neither ‘ A * nor
. ‘ ^ — A* is contradictory.)
5 6. A < B. 5 c. ‘ B * is an L-consequence
of ‘A*.
66. A = B. 6c. ‘A* and ‘B* are L-equi-
pollent (i.e. mutual L-conse-
quences).
76. C(A, B); 7c. ‘A* and ‘B* are L-com-
(A < ^ B). patible. B * is not an
L-consequence of ‘ A *.)
252
PART IV. GENERAL SYNTAX
Since the sentences given here are quasi-syntacticai, we can
interpret them as sentences either of the autonymous or of the
material mode of speech. In the case of the sentences of § 68, the
verbal formulations, or the verbal paraphrases given by the
authors, suggest interpretation in the autonymous mode of speech.
On the other hand, in the case of the symbolic sentences i ^ to 7
it is not clear which of the two interpretations is intended — in spite
of the fact that paraphrases (of the same kind as sentences i a to
7 a), and sometimes even detailed material explanations as well, are
given by the authors. In relation to a particular example, the
decisive question (as formulated in the material mode) is the fol-
lowing : Are ' I (A) * and ‘ A is impossible * to refer to the sentence
* A *, or to that which is designated by ‘ A * ? In the formal mode :
Is “A* is impossible* also to be a sentence? [If so, it must un-
doubtedly be equipollent to ‘A is impossible.*] If the answer is in
the affirmative, then ‘ 1 (A)* and ‘A is impossible* both belong to
the autonymous mode of speech; if in the negative, then they
belong to the material mode of speech. The authors do, it is true,
say that the sentences of modality are concerned with propositions,
but this assertion would decide the question only if it were quite
clear what was meant by the term ‘ proposition *. We will discuss
the two possibilities separately.
1. Suppose that by the term ‘proposition* the authors mean
what we mean by ‘sentence*. Then the term ‘proposition* is a
syntactical term' namely, the designation either of certain physical
objects in descriptive syntax or of certain expressional designs in
pure syntax. Then ‘A is impossible* is concerned with the sen-
tence ‘ A ’, hence is equipollent to ‘ ‘A* is impossible *, and belongs
to the autonymous mode of speech. In this case the intensionality of
the modal sentences does not depend upon the fact that they
speak about expressions (in the examples, about sentences, in
other cases, also about predicate-expressions) but upon the fact
that they do so according to the autonymous and not according
to the syntactical method.
2. Suppose that by a ‘ proposition * the authors mean not a sen-
tence (in our sense) but that which is designated by a sentence.
[For instance, in Lewis*s [Logic]^ pp. 472 ff., the distinction be-
tween ‘proposition* and ‘sentence* is possibly to be understood
in this way.] We will leave aside the question of what it is that is
§69. INTENSIONAL SENTENCES — LOGIC OF MODALITIES 253
designated by a sentence (some people say thoughts or the content
of thoughts, others, facts or possible facts); it is a question that
easily leads to philosophical pseudo-problems. So we shall simply
say neutrally “that which is designated by a sentence*’. In this
interpretation, the sentence ‘A is impossible* ascribes impossi-
bility not to the sentence ‘ A ’ but to the A which is designated by
the sentence. Here the impossibility is not a property of sen-
tences. “A* is impossible’ is not a sentence; it is therefore a
case not of the autonymous but of the material mode of speech.
‘ A is impossible * ascribes to the A which is designated by the sen-
tence a quasi-syntactical property, instead of to the sentence
‘A’ the correlated syntactical property (here ‘contradictory’).
[In this example, the second interpretation is perhaps the more
natural. It is the only possible one in the case of the formulation
‘ the process (or : state of affairs, condition) A is impossible ’ ; see
§ 79, Examples 33 to 35. On the other hand, we are perhaps more
inclined to relate a sentence about the consequence-relation or
about derivability to sentences rather than to that which is desig-
nated by them, and accordingly to choose the first interpretation. ]
We shall see later that, in general, the use of the material mode of
speech, though it is not inadmissible, brings with it the danger
of entanglement in obscurities and pseudo-problems that are
avoided by the application of the formal mode. So also here, the
systems of the logic of modalities are (on the whole) formally
correct. But if they are (in the accompanying text) interpreted in
the second way, that is, in the material mode of speech, then
pseudo-problems easily arise. This may perhaps explain the
strange and, in part, unintelligible questions and considerations
which are to be found in some treatises on the logic of modalities.
C. I. Lewis was the first to point out that in Russell’s language
[Princ, Math.] there is no way of expressing the fact that a certain
sentence necessarily holds or that a particular sentence is a conse-
quence of another. As against this, Russell can rightly maintain
that, in spite of it, his system is adequate for the construction both
of logic and of mathematics, that in it necessarily valid sentences can
be proved and a sentence which follows from another can be derived
from the former.
Although Lewis’s contention is correct, it does not exhibit any
lacuna within Russell’s language. The requirement that a language be
capable of expressing necessity, possibility, the consequence-relation,
etc., is in its^ justifiable ; it is fulfilled by us for instance in the case
254
PART IV. GENERAL SYNTAX
of our Languages I and II, not by means of anything supplementary
to these languages, but by the formulation of their syntax. On the
other hand, both Lewis and Russell — they are agreed on this point —
look upon the consequence-relation and implication as terms on the
same footing as relations between sentences, of which the first is the
narrower. For this reason, Lewis found himself obliged to extend
Russell’s language by introducing, in addition to Russell’s symbol of
implication ' D ’ (so-called material implication ; in our terminology :
proper implication), a new symbol * < ’ for what is called strict im^
plication (in our terminology: an intensional symbol of improper
implication without characteristic). This is intended to express the
consequence- relation (or derivability-relation), that is to say, in
I^ewis’s language, * A < B * is demonstrable if ‘ B * is a consequence of
‘A*. Lewis rightly pointed out that Russell’s implication does not
correspond to this interpretation, and that, moreover, none of the
so-called truth-functions (in our terminology: the extensional sen-
tential junctions) can express the consequence-relation at all. He
therefore believed himself compelled to introduce intensional sen-
tential junctions, namely, those of strict implication and of the
modality-terms. In this way his system of the logic of modalities
arose as an intensional extension of Russell’s language. The system
is set forth by Lewis in [Survey^^ pp. 291 ff., following MacColl, and
later presented in an improved form in [Logic'll pp. 122 fT., profiting
by the researches of Becker and others. To Russell’s system are added,
as new primitive symbols, symbols for ‘ possible ’ and ‘ strictly equi-
valent ’, and with the help of these, ‘ impossible ’, ‘ necessary ’, * strict
implication’, ‘compatible’, etc., are defined. Similar systems have
been constructed by Lewis’s pupils — by Parry ([/Co//.], p. 5), for ex-
ample, and Nelson ([//i/efwtona/]). Becker ([Moda/itd^s/ogiTt]), starling
out from Lewis’s [Survey\y has made some interesting investigations
using the same ifiethod. Before this Lukasiewicz had already worked
out so-called many-valued systems of the sentential calculus (see his
[Aussagenkalkiil]). In {Mehrwertige\ he interprets the sentences of
the three-valued calculus by a translation into the modal sentences ;
these are, as are Lewis’s, formulated in accordance with the quasi-
syntactical method.
It is important to note the fundamentally different nature of im-
plication and the consequence-relation. Materially expressed: the
consequence-relation is a relation between sentences; implication
is not a relation between sentences. [Whether, for example, Russell’s
opinion that it is a relation between propositions is erroneous or
not, depends upon what is to be understood by a “proposition”.
If we are going to speak at all of ‘ that which is designated by a sen-
tence then implication is a relation between what is so designated ;
but the consequence-relation is not. ] ‘ A 3 B ’ (S^) — as opposed to
§ 69. INTENSIONAL SENTENCES — LOGIC OF MODALITIES 255
the syntactical sentence '‘B* is a consequence of ‘A** (62) —
means, not something about the sentences and ‘B*, but, with
the help of these sentences and of the junction-symbol ‘ 0 \ some-
thing about the objects to which ‘A’ and ‘ B * refer. Formally ex-
pressed: ‘d’ is a symbol of the object-language, and ‘conse-
quence’ a predicate of the syntax-language. Of course, between
the two sentences Si and S2 there is an important connection (see
Theorem 14.7). S2 cannot, however, be inferred from Si but only
from the (equally syntactical) sentence ‘ Si is valid (or analytic)’.
The majority of the symbolic languages (for example, Russell’s
[Princ. Math.]) are (after a suitable extension of the rules of in-
ference) logical languages, and therefore contain no indeterminate
sentences. Hence, in these systems, S2 can be inferred from Si-
This explains why the sentences of implication are in general
erroneously interpreted as sentences about consequence-relations.
[This is one of the points which shows clearly how unfortunate it
is that the indeterminate sentences have, for the most part, been
disregarded in logical investigations.] The relation of the in-
tensional symbols of implication in the systems of the logic of modali-
ties, for instance that of the symbol of strict implication to ‘ D ’ and
to ‘consequence*, will become clear with the aid of the earlier
example on p. 235 ; this relation corresponds exactly to that sub-
sisting between ‘Limp’, ‘Imp’, and ‘consequence’. [We can
ignore here the differences between the intensional implications in
the various systems; they correspond to the different dehnitioiis
of the syntactical concept of ‘ consequence ’. ]
Russell’s choice of the designation ‘ implication ’ for the sentential
junction with the characteristic TFTT has turned out to be a very
unfortunate one. The words ‘to imply’ in the English language
mean the same as ‘ to contain ’ or ‘ to involve ’. Whether the choice
of the name was due to a confusion of implication with the con-
sequence-relation, I do not know; but, in any case, this nomen-
clature has been the cause of much confusion in the minds of many,
and it is even possible that it is to blame for the fact that a number of
j^eople, though aware of the difference between implication and the
consequence-relation, still think that the symbol of implication
ought really to express the consequence-relation, and count it as
a failure on the part of this symbol that it does not do so. If we have
retained the term ‘ implication ’ in our system, it is, of course, in a
sense entirely divorced from its original meaning; it serves in the
syntax merely as the designation of sentential junctions of a par-
ticular kind.
256
PART IV. GENERAL SYNTAX
§ 70. The Quasi-Syntactical and the Syntactical
Methods in the Logic of Modalities
All the foregoing systems of the logic of modalities (within the
province of modern logic, in symbolic language) have, it seems,
applied the quasi-syntactical method. This is not a matter of con-
scious choice between syntactical and quasi-syntactical methods;
rather the method applied is held to be the natural one. All in-
tensions! sentences of the previously existing systems of the logic
of modalities are, in any case, quasi-syntactical sentences, inde-
pendently of which of the two interpretations earlier discussed is
intended or (by a suitable incorporation in a more comprehensive
language) carried into effect. [Incidentally, it should be noted that
for" each of the systems one of the two interpretations can be
arbitrarily chosen and carried out, provided no attention is paid
to the authors* indications regarding interpretation. Accordingly,
it is, in particular, possible to interpret every sentence Si of the
logic of modalities that is intensional in respect of a partial ex-
pression 3 ti, in such a way that ?li is autonymous in Si-] Every
intensional system of the logic of modalities (and that even when
synthetic sentences are admitted as arguments) can be translated
into an extensional syntactical language, whereby every intensional
sentence, since it is quasi-syntactical, is translated into the corre-
lated syntactical sentence. In other words: syntax already con-
tains the whole of the logic of modalities, and the construction of
a special intensional logic of modalities is not required.
Whether, for the construction of a logic of modalities, the quasi-
syntactical or the syntactical method is chosen is solely a question
of expedience. We will not here decide the question but will only
state the properties of both methods. The use of the quasi-
syntactical method leads to intensional sentences, while the syn-
tactical method can also be carried into effect in an extensional
language. In a certain sense, the quasi-syntactical method is the
simpler ; and it may be that it will prove to be the appropriate one
for the solution of certain problems. It will only be possible to
pronounce judgment on its fruitfulness as a whole when the
method is further developed. Hitherto, if I am not mistaken, it has
in the main only been applied to the domain of the sentential
§ 70 . METHODS IN THE LOGIC OF MODALITIES 257
calculus which, on account of the resolubility of its sentences, is
quite a simple one (see Parry [^0//.], pp. 15 f.). It cannot be said
that the logic of modalities does not necessitate any syntactical
terms and is therefore simpler. For the construction of every
calculus, and therefor^ also of the logic of modalities, a syntax-
language is required in which the statement of the rules of in-
ference and of the primitive sentences is formulated (see §31); it
is usual simply to take the word-language for this purpose. Now,
as soon as this syntax-language is obtained, everything that it is
desired to express by the sentences o^ modality — and, in general,
far more — can be defined and formulated within it. That is the
reason why we have here given preference to the syntactical
method. It is, however, in any case, a worth-while task to develop
the quasi-syntactical method in general, and its use in the logic of
modalities in particular, and to investigate its possibilities in com-
parison with the syntactical method.
Even if in the construction of a logic of modalities we wish to use,
not the syntactical but the ordinary method hitherto employed,
the realization that this method is a quasi-syntactical one can help
us to overcome a number of uncertainties. These, for example,
have manifested themselves at various points in the fact that,
wishing to start from evident axioms, logicians have found them-
selves in doubt about the evidence of certain sentences ; it has even
happened that sentences which had previously been individually
regarded as evident have turned out later to be incompatible. As
soon, however, as it is seen that the concepts of modality — even
when they are formulated quasi-syntactically — are concerned with
syntactical properties, their relativity is recognized. They must
always be referred to a particular language (which may be other
than that in which they are formulated). In this way the problems
regarding the evident character of absolute relations between the
modality-concepts disappear.
§ 71. Is AN Intensional Logic Necessary.^
Some logicians take the view that the ordinary logic (for in-
stance, that of Russell) is deficient in some respects and must there-
fore be supplemented by a new logic, which is designated as in-
tensional logic or the logic of meaning (e.g. Lewis, Nelson
SL 18
PART IV. GENERAL SYNTAX
258
[Intensional], Weiss, and Jorgensen [Ziele], p. 93). Is this require-
ment justified? A close examination shows that two different
questions, which should be treated separately, are here involved.
1. Russell’s language is an extensional language. It is required
that it be supplemented' by an intensional language for the purpose
of expressing the concepts of modality (‘ consequence *, ‘ necessary *,
etc.). We have dealt with this question before, and have seen that
the concepts of modality may also be expressed in an extensional
language, and that their formulation only led to intensional sen-
tences because the quasi-syntactical method was used. Neither for
an object-language concerned with any domain of objects nor for
the syntax-language of any object-language is it necessary to go
outside the framework of an extensional language.
2. As opposed to the ordinary formal logic, a logic of content or
a logic of meaning is demanded. And, further, it is believed that
this second requirement also will be fulfilled by the construction
of an intensional logic of modalities ; thus it often happens that the
designations * intensional logic’ and ‘logic of meaning’ are used
synonymously. It is thought, that is, that the concepts of modality,
since they are not dependent merely upon the truth-values of
the arguments, are therefore dependent upon the meaning of the
arguments. This is often especially emphasized in connection with
the consequence-relation (e.g. Lewis [Survey], p. 328: “Inference
depends upon meaning, logical import, intension”). If all that is
meant by this is merely that, if the meanings of two sentences are
given, the question of whether one is a consequence of the other
or not is also determined, I will not dispute it (although I prefer to
regard the connection from the opposite direction, namely, the
relations of meaning between the sentences are given by means of
the rules of consequence ; see § 62). But the decisive point is the
following: in order to determine whether or not one sentence is a
consequence of another, no reference need be made to the meaning of
the sentences. The mere statement of the truth-values is certainly too
little; but the statement of the meaning is, on the other hand, too
much. It is sufficient that the syntactical design of the sentences be
given. All the efforts of logicians since Aristotle have been directed
to the formulation of the rules of inference as formal rules, that is to
say, as rules which refer only to the form of the sentences (for the
development of the formal character of logic, see Scholz [Ge-
§ 71 . IS AN INTENSIONAL LOGIC NECESSARY? 259
schichte\). It is theoretically possible to establish the logical relations
(consequence-relation, compatibility, etc.) between two sentences
written in Chinese without understanding their sense, provided
that the syntax of the Chinese language is given. (In practice this
is only possible in the^case of the simpler artificially constructed
languages.) The two requirements (i) and (2), which are usually
blended into one, arc entirely independent of one another. Whether
we wish to speak merely of the forms of the language or of the
sense (in some meaning of the word) of the sentences of S^, in
either case an intensional language may be used ; but we can also
use an extensional language for both these purposes. The differeiice
between the extensionality and intensionality of a language has no-
thing to do with the difference between the formal and the material
treatment. Now, is it the business of logic to be concerned with thg
sense of sentences at all (no matter whether they are given in ex-
tensional or in intensional languages)? To a certain extent, yes;
namely, in so far as the sense and relations of sense permil of being
formally represented. Thus, in the syntax, we have represented
the formal side of the sense of a sentence by means of the term
‘ content ’ ; and the formal side of the logical relations between sen-
tences by means of the terms ‘consequence*, ‘compatible’, and
the like. All the questions which it is desired to treat in the required
logic of meaning are nothing more than questions of syntax ; in the
majority of cases, this is only concealed by the use of the material
mode of speech (as is demonstrated by many examples in Part V).
Questions about something which is not formally representable,
such as the conceptual content of certain sentences, or the per-
ceptual content of certain expressions, do not belong to logic at all,
but to psychology. All questions in the field of logic can be for-
mally expressed and are then resolved into syntactical questions.
A special logic of meaning is superfluous' ‘non-formal logic* is a
contradictio in adjecto. Logic is syntax.
Sometimes the demand for an intensional logic is made in a third
connection : it is maintained that hitherto logic has only dealt with
the extension of concepts, whereas it should also deal with the in-
tension of concepts. But, actually, the newer systems of logic (Frege,
as early as 1893, followed by Russell and Hilbert) have got far beyond
the stage of development of the mere logic of extension in this sense.
Frege himself was the first to define in an exact way the old distinc-
tion between the intension and the extension of a concept (namely.
17-2
26 o
PART IV. GENERAL SYNTAX
by means of his distinction between a sentential function and its
course of values). One can rather maintain the reverse, that modem
logic, in its latest phase of development, has completely suppressed
extension in favour of intension (cf. the elimination of classes, § 38).
This misunderstanding has already been cleared up many times (see
Russell [Princ. Math.]^ i, p. 72; Carnap [Aufbau], p. 58, Scholz
[Geschichte]^ p. 63); it is always reappearing, however, amongst
philosophers who are not thoroughly acquainted with modem logic
(and amongst psychologists, who, in addition, confuse the logical and
the perceptual content of a concept).
(/) RELATIONAL THEORY AND AXIOMATICS
§ 71a. Relational Theory
In the theory of relations^ the properties of relations are in-
vestigated, particularly the structural properties — that is to
say those which are retained in isomorphic transformance. A
theory of this kind is nothing more than the syntax of many-
termed predicates. We have abandoned the usual distinction be-
tween the one-termed predicates and the class-symbols apper-
taining to them, and designate both class and property by pr^
(see §§ 37, 38). Similarly we no longer differentiate the w-termed
predicates for w.> i from the relational symbols which have hitherto
been correlated with them as symbols of extension. In this section,
we shall indicate briefly how the most important terms of the
theory of relations may be incorporated in the general syntax of
the predicates.
With regard to the terms used in the theory of relations (such
as ‘symmetrical*, ‘transitive*, ‘isomorphic*, etc.), it is important
to distinguish between their formulation in the object-language
and their formulation in the syntax-language. By means of this
distinction — the necessity of which is usually disregarded — certain
paradoxes in connection with the question of the multiplicity of
the transfinite cardinal numbers and the possibility of non-
denumerable aggregates are, as we shall see, clarified.
We will call an w-termed predicate homogeneous when, from a
sentence constructed from it and n arguments, another sentence
always arises as a result of any permutation of the arguments. The
majority of the terms of relational theory refer to homogeneous
two-termed predicates.
26i
§ 7 1 a . RELATIONAL THEORY
The relational properties of symmetry, reflexiveness, and so on
are expressed, according to the ordinary method introduced by
Russell, by means of predicates of the second level (or, in Russell’s
own symbolism, by class symbols of the second level). We will
write the deflnitions i^ the following form (employing the sym-
bolism of Language II, but leaving open the question as to whether
the expressions of the zero level are numerical expressions or
designations of objects) ;
(Fulfilment):* Erf (F) = (3 jc) (3 jy) (F(jc,y)) (i;
(Emptiness): Leer (F) = ^ Erf (F) (2)
(Symmetry):
Sym(F)= [Erf(F).(*)(3-) (FKy)DF(y,*))] (3)
(Asymmetry) : As (F) = (x) (y) (F(x,y) D ~F{y, x))
(Reflexiveness) :
Refl(F)= [Erf(F).(*)(3;)((F(*,j)VF(j,^c))DF(:r,A:))] (5)
(Total reflexiveness) :
Reflex(F)= [Erf(F).(A:) (F{x,x))] (6)
(Irreflexiveness) : Irr (F) = (x) F (:c, ;c)) (7)
(Transitivity):
Trans (F) = [(3 x) (3 j/) (3 z) (F{x,y),F(y, z)).
(x) (y) [z) (iF{x,y ) . F{y, z)pF{x, z)) ] (8)
(Intransitivity) :
Intr(F)H(jc)(j)(ir) [(F(jc,3').F(>',«))D~F(jc,a;)] (9)
We have altered the usual forms of the definitions (see Russell
[Princ. Math .] : Carnap [Logistik]) by introducing in the definiens
of (3), (5)1 (6), and ( 8 ) an existential sentence or ‘Erf(F)* as a
conjunction- term. According to the definitions hitherto given,
transitivity and intransitivity do not exclude one another ; and simi-
larly, neither do symmetry and asymmetry, reflexiveness and irre-
flexiveness. If, for instance, a relation has no intermediary term
(that is to say, no term which occurs in one pair of the relation as
second term, and in another pair as first term) then it is simul-
taneously both transitive and intransitive (because th& implicans in
the definiens of (9) is always false) ; and for the same reason a null
relation is at the same time transitive, intransitive, symmetrical,
asymmetrical, reflexive and irreflexive. On this account we intro-
duce conditions which require for symmetrical, reflexive, and transi-
* Erfulltheit.
262
PART IV. GENERAL SYNTAX
tive relations the propeity of non-emptiness, and further, for a
transitive relation, the occurrence of an intermediary term (non-
emptiness of the second power of the relation). On the basis of our
definition, the two terms of each of the three pairs exclude one
another. [The term * Erf (F) * in (6) can be left out if the individual
domain is non-empty, that is to say, if in the language in question,
' (3 x) (x = x)* is demonstrable, as is the case in the ordinary languages
of logistics.]
§ Jib. Syntactical Terms of Relational Theory
We will now introduce syntactical terms of relational theory as
opposed to the terms of relational theory of the object-language
which have been defined in the foregoing. The difference between
these two kinds of terms must be very carefully noted. Let us take
as an example the sentence ‘As(P)* — or, in the word-language:
‘ The relation P is an asymmetrical relation. * This sentence — we
will call it Si — is equipollent to the sentence
‘ (*) iy) [P {x,y) 0~?iy , *)] ’.
In contradistinction to this, we will say that the predicate ' P ’ (not
the relation P) is (systemically asymmetrical or) S -asymmetrical,
when Si is not merely true, but systemically true, i.e. valid; and
that * P * is (logically asymmetrical or) L-asymmetrical when Si is
(not merely valid but) analytic. In the material mode : The object-
sentence ‘ As (P) ’ or ‘ P is asymmetrical * expresses the fact that the
relation P does not hold in both directions in any pair; on the
other hand the syntactical sentence “ ‘ P * is S -asymmetrical means
that this fact can be inferred from the transformation rules of the
language-system S (hence, for example, from the natural laws, if
they are formulated as primitive sentences); and the syntactical
sentence “‘P* is L-asymmetrical” means that this is not a
genuinely synthetic fact, but is already determined by the L-rules
of S, and hence is given in substance by the definition of ‘P*.
We will formulate the definitions indicated here in a somewhat
different manner, so as to avoid the limiting assumption that
universal operators and symbols of proper negation and implica-
tion occur in the object-language S. The following are our
syntactical definitions. Let ptj be a homogeneous two-termed
predicate. [The definitions can easily be transferred to any
homogeneous Sfu^, and Sq^.j Then pti is called S-null
§ 71 ^. SYNTACTICAL TERMS OF RELATIONAL THEORY 263
(or L-null) if always (that is, here and in what follows, for any
closed arguments, pri(®i,9l2) is contravalid (or contra-
dictory, respectively), pti is called ^-fulfilled (or L-fulfilled,
respectively) when a valid (or analytic) sentence of the form
pTi (?!], 2I2) exists, pti^is called S-symmetrical (or L-symmetri-
cal) when pti is not S-null (or L-null, respectively) and pti (^2» ^1)
is always a consequence (or L-consequence) of pri(’jtti,%). pti is
called ^-asymmetrical (or h-asymmetricaly respectively) when
pri(3l2,9ti) and pri(?Ii,?l2) are always incompatible (or L-incom-
patible) with one another, pti is called ^-reflexive (or l^-reflexive)
when pti is not S-null (or L-null) and pti (?lii^i) is always a con-
sequence (or L-consequence) of pri(?ti,?l2) and always a conse-
quence (or L-consequence) of pri(?l2,?li); pti is called ^-totally
refleocive (or X^-totally reflexive) when ptj ('2fi, '‘2li) is always valid (or
analytic, respectively) ; pti is called S-irreflexive (or h-irreflexive)
when pri(?li,9Ii) is always contravalid (or contradictory, re-
spectively). pti is called ^-transitive (or \^-transitive) when the
two sentences pti(?Ii,^l2) and pti ('212, ^la) are not always incom-
patible (or L-incompatible, respectively) with one another, and
when pr2(?ti,'2l3) is always a consequence (or L-conscquencc, re-
spectively) of those two sentences; pt^is called '6-intransitive (or L-
intransitwe) when the above-mentioned three sentences are always
incompatible (or L-incompatible, respectively) with one another.
In the case of all these terms, corresponding V-ierms can be
defined ; pri is called P-null when pti is S-null but not L-null ; and
so forth.
We will again make clear the difference between the terms of
relational theory of the object-language and those of the syntax-
language by means of a juxtaposition.
The property of symmetry | The property of 6-symmetry
appertains to certain relations. appertains to certain predicates
(namely, to symbols of rela-
tions). (The same holds for
L-symmetry.)
This property is expressed by This property is expressed
the symbol ‘Sym’, or by the by the word ‘S-symmetrical’;
word ‘ symmetrical ’ ; these sym- this word belongs to the syntax-
hols belong to the object-lan- language,
guage.
PART IV. GENERAL SYNTAX
264
Assuming appropriate definitions for the predicates in a suitable
language S, the following examples hold. The predicate ‘brother’ is
L-irreflexive, but it is neither S-symmetrical nor S-asymmetrical.
If it follows from the rules of transformation of S that, in the district
B, at least one man has a brother but no man has a sister, then
‘brother in B’ is S-symmetrical, but not L-symrnetrical, and is
therefore P-symmetrical. ‘ Father ’ is L-irreflexive, L-asymmetrical
and L-intransitive.
Theorem yib.i. (a) If the predicate ‘P* is L-symmetrical or
P-symmetrical, then it is also S-symmetrical. (^) If ‘P* is S-
symmetrical, then P (not ‘P*) is symmetrical; the converse is not
universally true, (c) Let S be an L-language (which may also be a
descriptive language like I and II); then if ‘P* is L-symmetrical in
S, it is also S-symmetrical ; and conversely, (d) Let S be a logical
language (hence an L-language); then if ‘P* is S-symmetrical or
L-sfym metrical in S, P is symmetrical; and conversely. Corre-
sponding theorems are true for the remaining terms. For and
id, it is assumed that the language S contains its own syntax; S is
here taken as a word-language, in which ‘P is symmetrical* is
written for ‘ Sym (P) *.
It would be equally possible to express the syntactical terms here
defined by means of second level predicates of the object-language —
for example: “P’ is S-irreflexive ’ by ‘Slrr(P)’ and “P* is L-
irreflexive ’ by ‘ LIrr (P) ’. But in ‘ Slrr (P) ’ and ‘ LIrr (P) ‘ P * would
be autonymous, which is not the case in ‘Irr(P)* (in so far as de-
scriptive arguments are admitted; see p. 237). Those sentences are
quasi-syntacticaly but ‘ Irr(P)* is not (see Example i on p. 234).
§ 71C, Isomorphism
We will define a few more terms of relational theory leading up
to the particularly important term ‘ isomorphism *. First we will
give, as before, definitions of symbols of an object-language (with
a symbolism like that of Language II).
(Converse) : cnv (F) {x^y) = (F(>', a?)) (i)
(One-many) :
Un(i?) = (a:)(jy)(3:) [(i?(*,2r).F(>',2:))3(a;=j>')] (2)
(One-one) : Unun (F) = (Un (F) • Un [cnv (F)] ) (3)
§7IC. ISOMORPHISM
265
(Correlator):
Korr (Hf G) = (Unun (//) • («) [(3 v) (F (m, v) V F(z;, m)) =
(3 •») {H («. x ))] . (jc) [(3 y) (G (*, y)'^G ac))s (3 u)
(H («, *))] . (tt) (») {x) (y) [(H {u, x) .H{v,y))o (F(u, t)) =
G(^,3'))]) . (4)
(Isomorphism):
Is{F,G) = (2H)(Kott{H,F,G)) ( 5 )
These definitions correspond (in a somewhat different formula-
tion) to the usual ones. (4) is here formulated for two-termed
predicates, but can easily be transferred to w-termed predicates
for w > 2. Just as, earlier, we opposed the terms of relational
theory of the object-language (such as ‘Irr*) to corresponding
syntactical terms (such as ‘ S-irreflexive* and ‘ L-irreflexive *),
so here also we must contrast the terms of the object- language
that are defined in (i) to (5) with syntactical terms that have
previously been either ignored or confused with the former.
Let pti be a homogeneous two-termed predicate (the definitions
can easily be transferred to 'JJr, Sfu, or Sg). pr2 is called the
^-converse of pti if always (that is to say, here and in the following,
for any closed arguments) pr2(^i,9X2) is equipollent to pri(?l2>^i)-
pti is called ^-one-many if 5ti and always synonymous in rela-
tion to {pri(3li,®3),pri(9l2,3l3)}‘ pri is called S-one-one if prj and
the S -converse of pti are S -one-many. Let pri and pr2 be homo-
geneous w-termed predicates; then pta is called an ^-correlator for
pri and pr2 if the following conditions are fulfilled: (i) pta is
S-one-one; (2) if is a suitable argument for pri then it is also
a suitable argument of the first place for pra, and conversely;
(3) if 2ti is a suitable argument for pta then it is also a suit-
able argument of the second place for pta, and conversely; (4)
P^I (^1, • • • ^n) and pta ( 3 li', . . . ?!„') are always equipollent in
relation to (ptj (?ti, ^l/), pta (^la, Sta'). • • • Pts ( 5 tn. %.')}• Two homo-
geneous w-termed predicates, pti and ptg, are called S-isomorphic
if there is an S-correlator of pti and pta- For each one of these
terms there is to be defined an analogous L-term and P-term.
Theorem 7IC.1. Let the language S contain its own syntax.
[Here we will take a word-language and will write “P and Q are
isomorphic” (not ‘P’ and ‘Q') instead of ‘Is(P, Q)’.] Then
(analogously to Theorem ^la.ih and i d) it is true that : if ‘ P * and
266
PART IV. GENERAL SYNTAX
‘Q* are S- (or L-) isomorphic, then P and Q are isomorphic; it
S is a logical language, then the converse is also true.
An S-correlator for pr^ and pr2 is a predicate of the object-
language. As distinguished from this, we mean by a syntactical
correlation of two homogeneous w-termed predicates, pti and pr2,
a one-one syntactical correlation, which fulfils the following
conditions: (i) if is a suitable argument for pti, then is
a suitable argument for pr2; (2) if is a suitable argument
for ptg then there is a suitable argument for ptj such that
QiPy is (3) P^iC^Ii» 5 l 2 ) is always equipollent to pr2(Qi ['ili],
Qi['M 2]). Two homogeneous w-termed predicates, ptj and pr2,
are called syntactically isomorphic when there is a syntactical
correlation for them (that is to say, when such a correlation can
be defined In the syntax-language, assuming it to be sufficiently
rich).
We will make the difference between the concepts of isomor-
phism quite clear by means of a contrasting table ; this is analogous
to the earlier one, but here a third kind of concept, namely, syn-
tactical isomorphism, is introduced.
1. The relation of ^-iso-
morphism subsists between cer-
tain (homogeneous, two- or
many-termed) predicates (name-
ly, symbols of relations). (The
same holds for L-isornorphism.)
This relation is expressed by
the word ‘ S-isomorphic * ; this
word belongs to the syntax-
la 7 igua(fe,
2. The relation of syntactical
isomorphism likewise subsists
between certain predicates. It
is expressed by the words ‘ syn-
tactically isomorphic’; these
words belong to the syntax-
language.
S-isomorphism and syntactical isomorphism are thus both
syntactical concepts which refer to predicates of the object-
The relation of isomorphism
subsists between certain (homo-
geneous, two- or many-termed)
relations.
This relation is expressed by
the symbol ‘ Is or by the word
‘ isomorphic ’ ; these symbols
belong to the object-language.
§ 7 It. ISOMORPHISM 267
language. The difference between the two concepts consists in the
fact that in S-isomorphism the one-one correlation is brought
about by means of a predicate of the object-language, and in syn-
tactical isomorphism, on the other hand, by any syntactical terms.
Thus it may happen jhat two predicates, although they are syn-
tactically isomorphic, are not S-isomorphic ; namely, when the
object-language contains no suitable correlator. Since the
majority of mathematical calculi (when their rules of transforma-
tion have, if necessary, been suitably completed) contain only
logical symbols, in their case, in accordance with Theorem (i),
isomorphism (‘Is’) and S-isomorphism coincide. [To be more
exact, they appertain to corresponding pairs: isomorphism to a
pair of relations, S-isomorphism to the corresponding pair of
predicates. Formally expressed: ‘Is* is in this case quasi-sya-
tactical; ‘S-isomorphic* is the correlated syntactical predicate.]
But even here, the difference between S-isomorphism and syn-
tactical isomorphism must be noted.
Theorem 710.2. If two predicates are S- (or L-) isomorphic,
then they are also syntactically isomorphic. The converse is not
universally true (even if S is a logical language).
§ yirf. The Non-Denumerable Cardinal Numbers
If due attention is paid to the difference between S-isomorphism
and syntactical isomorphism, certain paradoxes in connection with
the Theory of Aggregates can be explained. We may consider as
an example the theorem of the multiplicity of transfinite cardinal
numbers, which is one of the main supports of the Theory of
Aggregates. The one-termed predicates are designations of ag-
gregates; the isomorphism of two such predicates corresponds to
equality of their cardinal numbers (‘similarity* or ‘equivalence*
in the terminology of the Theory of Aggregates). Let us take as
object-language S the system of axioms used in Fraenkers Theory
of Aggregates {\Mengenlehre^, § supplemented by a sentential
and a functional calculus (in the word-language). The theorem
that more than one transfinite cardinal number exists depends
upon the theorem that the aggregate U (M) of the sub-aggregates
of an aggregate M has a higher cardinal number than has M ; this
theorem is based upon what is known as Cantor*s theorem, which
268
PART IV. GENERAL SYNTAX
maintains that M and U(M) cannot have the same cardinal
number. Fraenkel [Untersuchungen] has given a proof of this
thecjrem which remains valid for his system S even though it con-
tains the so-called Axiom of Limitation {[Mengenlehre], p. 355).
On the other hand, however, we arrive at a contrary result as a conse-
quence of the following argument. The Axiom of Limitation means
that in the aggregate-domain which is treated in S — let us call it
B — only those aggregates occur of which the existence is required
by the other axioms. Therefore, only the following aggregates are
existent in B ; in the first place, two initial aggregates, namely, the
null-aggregate and the denumerably infinite aggregate, Z, re-
quired by Axiom VII; and secondly, those aggregates which can
be constructed on the basis of these initial aggregates by applying
an arbitrary but finite number of times certain constructional
procedures. There are only six kinds of these constructional steps
(namely, the formation of the pair-aggregate, of the sum-
aggregate", of the aggregate of sub-aggregates, of the aggregate of
Aussonderungy of the aggregate of selection, and of the aggregate
of replacement). Since only a denumerable multiplicity of aggre-
gates can be constructed in this way, there is in B, according to the
Axiom of Limitation, only a denumerable multiplicity of aggre-
gates, ami consequently, at the most, only a denumerable multi-
plicity of sub-aggregates of Z. Therefore U (Z) cannot have a
higher cardinal number than Z. Actually, on the basis of the two
initial aggregates and the six constructional steps, it is easy to
give a method of denumerating all the aggregates of B, and hence
also of the sub -aggregates of Z, and in this way the sub -aggregates
of Z can be univocally correlated with the elements of Z. Therefore
U (Z) and Z have the same cardinal number.
This result appears to contradict Cantor’s theorem; but the
contradiction disappears as soon as we differentiate betzveen equality
of cardinal numbers and syntactical equality of cardinal numbers.
[Since S is a logical language, equality of cardinal numbers and
S-equality of cardinal numbers coincide. ] According to Fraenkel’s
definition {[Mengenlehrely p. 314) two (mutually exclusive)
aggregates M and N have the same cardinal number only if (in B)
there is a transforming aggregate (i.e. a correlator) Q — that is, an
aggregate of mutually exclusive pairs (w, where m is an element
of M and w of N such that the pairs exhaust M and N. Now if
§ 71 ^/. the non-denumerable cardinal numbers 269
M is denumerably infinite, a one-one correlation of the kind
mentioned before can be effected between the elements of M and
those of U (M), and hence between the elements and the sub-
aggregates of M. This correlation, however, is not a correlator
in S but a syntactical, correlation. In B there is no aggregate Q
which could be a correlator for M and U (M) ; that is shown by
Fraenkel’s proof. But now Fraenkel’s proof and our own findings
are no longer in contradiction with one another : M and U (M),
although they have dijf event cardinal numbers are nevertheless syn~
tactically of the same cardinal number.
In syntax it is always possible to effect a denumeration of ex-
pressions of any kind (in an arithmetized syntax, for instance, by
means of the series-numbers of the expressions). Thus in relation
to a fixed syntax-language (which must be presupposed for the
construction of the system S) every aggregate of FraenkeVs domain
of aggregates B is syntactically denumerable; two transfinite aggre-
gates are always syntactically of the same cardinal number. This is
the element of truth in the criticism brought by the Intuitionists
against the concept of the non-denumerable aggregates. [Poincare
{[Gedanken]^ pp. 108 ff., 134 ff.) bases his rejection of the non-
denumerably infinite — subsequently maintained by Brouwer [In-
tuitionism] and others — on this nominalistic view, which he him-
self, not very happily, designates as idealistic. ] It must, however,
be noted that the syntactical equivalence of all transfinite aggre-
gates of B (from the standpoint of a fixed syntax-language) is not
in contradiction with their non-equivalence (within the system S),
and that therefore the distinction between different transfinite
cardinal numbers within a system of the Theory of Aggregates is
justified. And indeed, in FraenkePs system of axioms, which,
because of the Axiom of Limitation, is, in a broad sense, a con-
structive system, the inequivalence of certain aggregates — for
instance that of Z and U (Z) — follows from a certain poverty of the
system : it docs not contain any aggregate which in the given cases
could serve as a correlator. In non-constructive axiom-systems —
for instance, in a system which contains no Axiom of Limitation,
and which, on the other hand, operates with existential axioms to
greater extent — the inequivalence, say, of M and U (M) can be
attributed, conversely, to a certain richness of the system : U (M)
contains so many element-aggregates that they cannot be corre-
270
PART IV. GENERAL SYNTAX
lated in a one-one correspondence with the elements of M. Of
course, this does not mean that such a wealth of aggregate-rf^w^na-
tions exists within the system ; obviously the number of aggregate-
designations is denumerable in every system. The richness is only
assumed by means of axioms, and is not demonstrable by designa-
tions (names or descriptions).
Further it must be noted that the difference between the aggre-
gates of the natural numbers, of the real numbers, of the functions
of real numbers, and so on, which Cantor has pointed out and
formulated by attributing to them different cardinal numbers, is
also syntactically representable. This distinction is particularly
significant for the syntactical investigation of a series of languages
each of which is contained in the next as a proper sub-language.
That characteristic of the class of the logical numerical functors
which Cantor designates as the non-denumerability of the aggre-
gate of the real numbers is expressed, for instance, in an in-
creasing series of languages by the fact that every language of the
series, in addition to the denumerably many such functors of the
previous languages, can always contain new ones (on this point,
see our earlier remarks on the diagonal method, on Richard’s
antinomy, and on the defectiveness of arithmetic; compare
Theorems 60C.3 and 6o</.i),
As a result of the distinction between denumerability (in the
system under consideration) and syntactical denumerability, the
paradox in connection with the famous Lowenheim-Skolem
theorem (Skolem [Erfullbarkeit]; cf. Fraenkel [Mengenlehre]^ p.
333) also disappears. This theorem means approximately that for
a non-contradictory axiom-system S of the Theory of Aggregates
there is always already a model in a denumerable domain. Such a
model, however, is not constructed by means of terms of S, but by
means of discussions about S, that is to say, by means of syntactical
terms. And the denumerability of the domain whose elements
constitute the model is not demonstrated by the production of a
correlator in S, but by the proof of the constructibility of a syn-
tactical correlation. It is, accordingly, not the denumerability
(in S) of a model which is proved, but only the syntactical de-
numerability. Thus the Skolem theorem does not contradict
Cantor’s theorem (or Fraenkel’s proof).
§ 71 ^. the axiomatic method
271
§ yie. The Axiomatic Method
An axiom-system (abbreviation ‘ AS *) is usually regarded as a
system of sentences, the so-called axioms, from which other
sentences, the so-called theorems or conclusions, may be deduced.
The axioms consist partly of symbols whose meaning is assumed
to be known already (for the most part, logical symbols), and partly
of symbols which are introduced for the first time by the AS, the
so-called primitive symbols of the AS. It is customarily said that
no meaning is presupposed for the latter, but, that the AS — as a
sort of implicit definition — determines their meaning. In order to
draw conclusions from the axioms, obviously the rules of forma-
tion and transformation of the language concerned must be known.
These rules are usually tacitly assumed, but in an exact formulation
of the AS this tacit assumption must be replaced by an explicit
statement. Further, it is characteristic of the axiomatit method
that the primitive symbols are, to a certain extent, determined by
the AS only in relation to one another. Hence there is sometimes
the possibility of interpreting the primitive symbols in several
different ways. The statement of a certain interpretation of the
primitive symbols is designated as the establishment of corre-
lative definitions (see p. 78). If it is proved that the axioms are
fulfilled for a certain interpretation, or at least that their fulfilment
is not excluded, we say that by this interpretation a model for the
AS is constructed.
Example: In drawing up an AS of Geometry, it is usual merely to
state the specifically geometrical axioms. In order to render de-
ductions possible, the sentential and functional calculus, together
with elementary arithmetic, must be added.
Usually the AS is formulated in the word-language without any
precise statement of the syntactical rules, particularly the rules of in-
ference. Now there are several quite different possibilities of putting
such an AS into the exact form of a calculus. We will state briefly
the most important methods of formulation. It is desirable to
choose a different terminology for each of the three methods, so
that it may always be clear which one is the subject of discussion.
Therefore we shall speak of “ axioms** only in connection with the
first method, of “primitive sentences** in connection with the
272
PART IV. GENERAL SYNTA".
second (in accordance with our regular usage in this book), and
of “ premisses*' in connection with the third.
First method: the axioms as sentential functions.
For the representation of the AS, a language S with a sentential
and a functional calculus will be taken. (For the examples in the
following, we shall use the symbolism of Language II.) Each of
the k primitive symbols of the AS is represented by a o (or ®) ;
we call these 0 xht primitive variables. Each of the m axioms is then
formulated as an Gfu, and, specifically, as an Gfu* if the axiom
contains i different primitive symbols. The same holds of the con-
clusions. In the deductions, however, there is no substitution for
the free primitive variables. (In the material mode of speech: the
primitive variables do not express universality, but indeterminate-
ness.) Sfu^ is called a conclusion from the m axioms SfUi, ...
Sfu„j, if the universal implication-sentence
(Oi) . . . (Ot) [(SfUl . Sfu D Sfu„]
is analytic (or L''-demonstrable) in S. According to this method,
a model for the AS is to be understood as a series of k substitution-
values 3Ii, . . . 31* for the primitive variables. If
is valid (or not contravalid, or not contradictory, respectively) in
S, then the model is called a real (or a really possible, or a logically
possible) model. If at least one of the substitution-values is
descriptive, then the model is called descriptive ; otherwise, logical
(or mathematical).
The advantage of this method consists in the fact that by it a
common language may be used for all AS*s, and for all AS*s of the
usual kind a simple language of the usual kind having a sentential
and a functional calculus. The primitive variables in this con-
nection are usually or p ; in the ordinary AS only ®d, and ^p
occur, and for the most part ^p.
Example: If Hilbert’s AS of Euclidean geometry {[Grundl. Geom.],
p. i) is presented in accordance with the first method, seven dif-
ferent primitive variables appear: ‘point’, ‘straight line*, ‘plane*
will each be represented by a ‘lies upon*, by a ‘between*.
§ 71 ^. THE AXIOMATIC METHOD 273
by a ; and ‘ congruence of segments * and ‘ congruence of angles *
each by a
On the first method, see Carnap {{Eigentliche^^ \Logistik'\y pp. 71 ff.,
[Asciomatik}).
Second method: the axioms as primitive sentences.
The axioms of the AS are formulated as the. primitive sentences
of a language S^. Sometimes, in this case, the axioms of a given
AS are the only primitive sentences of Sj, so that only rules of
inference have to be added. But sometimes not only the rules of
inference but also the L-primitive sentences of S^ are tacitly
assumed, so that the given axioms must be formulated as additional
primitive sentences of S^ (for the most part descriptive P-sentences).
The conclusions of the AS are the sentences that are valid (or
demonstrable) in S^. The primitive symbols of the AS are here
primitive symbols of S^; and either they are the only primitive
symbols of S^ or they are additional primitive symbols (mostly
descriptive) added to the original logical primitive symbols of S^
(which in the ordinary formulation of the AS are tacitly assumed).
The primitive symbols are not 35 . Hence, the construction of a
model can here not be effected by substitution. It is achieved by
means of a translation, ^1 another language Sg (usually
a language of science which has a practical use). In the majority of
cases this will be an expressional translation ; the statement of the
model consists, as a rule, only uf the statement of the Qj-correlates
of the additional primitive symbols, the translation of the logical
primitive symbols being assumed to be established and well
known. The model is said to be real (or really possible, or logically
possible) if the class of the -correlates of the axioms of the AS is
valid (or not contravalid, or not contradictory, respectively) in S.2.
If this class is descriptive, the model is called descriptive; if it is
logical, the model is called logical (or mathematical).
Example: On a system of geometrical axioms in accordance with
the second method, see §25 II A ‘‘Axiomatic Geometry”; arith-
metical geometry (I) constitutes a logical model, physical geometry
(II b) a descriptive model.
The second method affords a greater freedom in interpretation,
and thus in the construction of models, than the first. In the first
method, the domain of the interpretations of a certain primitive
symbol is the domain of the substitution-values of the primitive
SL
19
274 part IV. GENERAL SYNTAX
variable. If, as is usual, it is a case of primitive variables within a
system of types, then the same relations of types must hold be-
tween the symbols of the model as hold between the corresponding
primitive variables. In the second method, the place of substitu-
tion is taken by the far more elastic operation of translation ; here,
for instance, isogenous primitive symbols can have correlates
which are not isogenous.
Examples: i. Let Si, S2, and S3 be presentations of AS*s of Euclid-
ean geometry in accordance with the second method. Let Si take
straight lines as classes of points (see Carnap \LogistiK\y § 34); let Sg
take straight lines as relations between points (see Carnap \Logistik\y
§ 35) ; and let S3 take straight lines and points as individuals (as does
Hilbert [Grundl. Geom.]). Three AS*s of this kind, formulated in
accordance with the first method, cannot have a common model. On
the other hand, by the second method this is possible, in the sense
that Si, S2, and S3 can all be translated into the same sub-language
of a logical language, in which a point is interpreted in the usual way
as a triad of real numbers, a plane as a class of such triads which
satisfy a linear equation, and so on. Thus, by this method, it is easy
to portray formally the relationship of the three AS’s, which is what
is meant when it is said that they represent the same geometry.
2. Let an AS of the Theory of Aggregates be given which takes all
aggregates as individuals (as, for instance, Fraenkel does [Mengen-
lehre]^ § 16) but in which only homogeneous aggregates occur (30
that, for example, as opposed to the AS of Fraenkel, m and {m}
cannot be elements of the same aggregate). If an AS of such a kind
is presented in accordance with the second method, it can be in-
terpreted as a theory of classes, and, in spite of the equal level of the
aggregates, as a theory of classes of all levels. ®a and certain of
all levels (for instance in Language II) are taken as correlates of the
aggregate-expressions.
Third method: the axioms as premisses.
The AS is represented by means of a (usually indeterminate)
sentential class of an assumed language S. The conclusions are
here the L-consequences of this class, and hence the axioms appear
as premisses of derivations (or of consequence-relations). In this, as
in the second method, an interpretation consists of a translation ;
and, as in the first method, it is possible to formulate several AS’s
within the same language.
Special and general axiomatics, that is, the theory of certain
individual AS’s or of AS’s in general, is nothing more than the
syntax of the AS's, The investigations in axiomatics, which have
275
§71 ^. the axiomatic method
been conducted chiefly and intensively by mathematicians, thus
contain a great number of syntactical discussions and definitions,
many of which we have already been able to apply in this outline
of a general syntax. We have defined some terms, in accordance
with the second method, as properties of languages, and some (in
part, the same ones),* in accordance with the third method, as
properties of sentential classes. [For instance, the terms ‘re-
futable*, ‘ L-refutable ’, ‘ contravalid *, and ‘contradictory’, which
refer to sentential classes, correspond to the terms ‘contradictory*,
‘L-contradictory*, ‘inconsistent*, and ‘L-inconsistent’, which
refer to languages. ] Conversely, it will be possible to make use of
the findings and definitions of general syntax for axiomatics. But
we cannot go more fully into this subject in the present work.
Full bibliographical references on the subject of axiomatics t«>
the year 1928 arc given by Fraenkel [Menfienlehre], § 18. Some new
works on the subject are as follows : Hertz [AxiQm,\, Lewis and Lang-
ford [Logic]f and Tarski [Methodologie]^ [Widerspruchsfr.y
PART V
PHILOSOPHY AND SYNTAX
A. ON THE FORM OF THE SENTENCES
BELONGING TO THE LOGIC OF SCIENCE
§ 72. Philosophy Replaced by the Logic
OF Science
The questions dealt with in any theoretical field — and similarly
the corresponding sentences and assertions — can be roughly
divided into object-questions and logical questions, (This differentia-
tion has no claim to exactitude; it only serves as a preliminary to
the following non-formal and inexact discussion.) Qy object-
questions are to be understood those that have to do with the
objects of the domain under consideration, such as inquiries re-
garding their properties and relations. The logical questions, on
the other hand, do not refer directly to the objects, but to sen-
tences, terms, theories, and so on, which themselves refer to the
objects. (Logical questions may be concerned either with the
meaning and content of the sentences, terms, etc., or only with the
form of these ; of this we shall say more later.) In a certain sense,
of course, logical questions are also object-questions, since they
refer to certain objects — namely, to terms, sentences, and so on —
that is to say, to objects of logic. When, however, we are talking of
a non-logical, proper object-domain, the differentiation between
object-questions and logical questions is quite clear. For instance,
in the domain of zoology, the object-questions are concerned with
the properties of animals, the relations of animals to one another
and to other objects, etc. ; the logical questions, on the other hand,
are concerned with the sentences of zoology and the logical con-
nections between them, the logical character of the definitions
occurring in that science, the logical character of the theories and
hypotheses which may be, or have actually been, advanced, and
so on.
According to traditional usage, the name ‘philosophy* serves
as a collective designation for inquiries of very different kinds.
278 PART V. PHILOSOPHY AND SYNTAX
Object-questions as well as logical questions are to be found
amongst these inquiries. The object-questions are in part con-
cerned with supposititious objects which are not to be found in the
object-domains of the sciences (for instance, the thing-in-itself,
the absolute, the transcendental, the objective idea, the ultimate
cause of the world, non-being, and such things as values, absolute
norms, the categorical imperative, and so on); this is especially
the case in that branch of philosophy usually known as meta-
physics. On the other hand, the object-questions of philosophy
are also concerned with things which likewise occur in the em-
pirical sciences (such as mankind, society, language, history,
economics, nature, space and time, causality, etc.); this is especi-
ally the case in those branches that are called natural philosophy,
tne philosophy of history, the philosophy of language, and so on.
I'he logical questions occur principally in logic (including applied
logic), and also in the so-called theory of knowledge (or epistemo-
logy), where they are, however, for the most part, entangled with
psychological questions. The problems of the so-called philo-
sophical foundations of the various sciences (such as physics,
biology, psychology, and history) include both object-questions
and logical questions.
The logical analysis of philosophical problems shows them to
vary greatly in character. As regards those object-questions whose
objects do npt occur in the exact sciences, critical analysis has re-
vealed that they are pseudo-problems. The supposititious sen-
tences of metaphysics, of the philosophy of values, of ethics (in so
far as it is treated as a normative discipline and not as a psycho-
sociological investigation of facts) arc pseudo-sentences ; they have
no logical content, but are only expressions of feeling which in
their turn stimulate feelings and volitional tendencies on the part
of the hearer. In the other departments of philosophy the psycho-
logical questions must first of all be eliminated; these belong to
psychology, which is one of the empirical sciences, arid are to be
handled by it with the aid of its empirical methods. [By this, of
course, no veto is put upon the discussion of psychological ques-
tions within the domain of logical investigation; everyone is at
liberty to combine his questions in the way which seems to him
most fruitful* It is only intended as a warning against the dis-
regard of the difference between proper logical (or epistemological)
§72. PHILOSOPHY REPLACED BY THE LOGIC OF SCIENCE 279
questions and psychological ones. Very often the formulation of a
question does not make it clear whether it is intended as a psycho-
logical or a logical one, and in this way a great deal of confusion
arises. ] The remaining questions, that is, in ordinary terminology,
questions of logic, of ^e theory of knowledge (or epistemology), of
natural philosophy, of the philosophy of history, etc., are some-
times designated by those who regard metaphysics as unscientific
as questions of scientific philosophy. As usually formulated, these
questions are in part logical questions, but in part also object-
questions which refer to the objects of the special sciences. Philo-
sophical questions, however, according to the view of philosophers,
are supposed to examine such objects as are also investigated by
the special sciences from quite a different standpoint, namely,
from the purely philosophical one. As opposed to this, we shall
here maintain that all these remaining philosophical questions are
logical questions. Even the supposititious object-questions are
logical questions in a misleading guise. The supposed peculiarly
philosophical point of view from which the objects of science are
to be investigated proves to be illusory, just as, previously, the
supposed peculiarly philosophical realm of objects proper to meta-
physics disappeared under analysis. Apart from the questions of
the individual sciences, only the questions of the logical analysis of
science, of its sentences, terms, concepts, theories, etc., are left as
genuine scientific questions. We shall call this complex of ques-
tions the lo^ic of science. [We shall not here employ the expression
‘theory of science’; if it is to be used at all, it is more appropriate
to the wider domain of questions which, in addition to the logic
of science, includes also the empirical investigation of scientific
activity, such as historical, sociological, and, above all, psycho-
logical inquiries.]
According to this view, then, once philosophy is purified of all
unscientific elements, only the logic of science remains. In the
majority of philosophical investigations, however, a sharp division
into scientific and unscientific elements is quite impossible, hor
this reason we prefer to say : the logic of science takes the place of the
inextricable tangle of problems zvhich is known as philosophy. Whether,
on this view, it is desirable to apply the term ‘philosophy* or
‘scientific philosophy’ to this remainder, is a question of ex-
pedience which cannot be decided here. It must be taken into
PART V. PHILOSOPHY AND SYNTAX
280
consideration that the word ‘philosophy* is already heavily
burdened, and that it is largely applied (particularly in the German
language) to speculative metaphysical discussions. The designation
‘ theory of knowledge * (or ‘ epistemology *) is a more neutral one,
but even this appears not to be quite unobjectionable, since it mis-
leadingly suggests a resemblance between the problems of our
logic of science and the problems of traditional epistemology ; the
latter, however, are always permeated by pseudo-concepts and
pseudo-questions, and frequently in such a way that their dis-
entanglement is impossible.
The view that, as soon as claims to scientific qualifications are
made, all that remains of philosophy is the logic of science, cannot
be established here and will not be assumed in what follows. In
this part of the book we propose to examine the character of the
sentences of the logic of science, and to show that they are syn-
tactical sentences. For anyone who shares with us the anti-
metaphysical standpoint it will thereby be shown that all philo-
sophical problems which have any meaning belong to syntax. The
following investigations concerning the logic of science as syntax
are not, however, dependent upon an adherence to this view;
those who do not subscribe to it can formulate our results simply
as a statement that the problems of that part of philosophy which
is neither metaphysical nor concerned with values and norms are
syntactical.
Anti-metaphysical views have often been put forward in the past,
especially by Hume and the Positivists. The more exact thesis that
philosophy can be nothing other than a logical analysis of scientific
concepts and sentences (ifi other words, what we shall call the logic
of science) is represented in particular by Wittgenstein and the
Vienna Circle, and has been both established in detail and in-
vestigated in all its consequences by them ; see Schlick [Metaphysik]^
[Wende]^ \Positivismus^\ Frank \Kau 5 algesetz\\ Hahn \Wiss.
Weltauff.]; Neurath \Wiss. Weltauff.], [Wege]; Carnap [Meta-
physik]; further bibliographical references are given by Neurath
[IFws. Weltauff.] and in ErkenntniSy i, 315 ff. Neurath is definitely
opposed to the continued use of the expressions ‘philosophy*,
* scientific philosophy *, ‘ natural philosophy *, ‘ theory of knowledge *,
etc.
The term ‘ logic of science * will be understood by us in a very wide
sense, namely, as meaning the domain of all the questions which are
usually designated as. pure and applied logic, as the logical analysis
of the special sciences or of science as a whole, as epistemology, as
§ 72. PHILOSOPHY REPLACED BY THE LOGIC OF SCIENCE 28 1
problems of foundations, and the like (in so far as these questions
are free from metaphysics and from all reference to norms, values,
transcendentals, etc.). To give a concrete illustration we assign the
following investigations (with very few exceptions) to the logic of
science : the works of Russell, Hilbert, Brouwer, and their pupils, the
works of the Warsaw logicians, of the Harvard logicians, of Reichen-
bach’s Circle, of the Vienna Circle centring around Schlick, the
majority of the works cited in the bibliography of this book (and
others by the same authors), the articles in the journals Erkenntnis
and Philosophy of Science, the books in the collections “ Schriften
zur wissenschaftlichen Weltauffassung** (edited by Schlick and
Frank), “ Einheitswissenschaft** (edited by Neurath), and finally
the works mentioned in the following bibliographies: Erketmtnis,
I, 315 ff. (general), 335 ff. (Polish logicians); ii, 15 1 ff. (foundations
of mathematics), 189 f. (causality and probability); v, 185 ff.
(general), 195 ff. (American authors), 199 ff. (Polish authors), 409 ff.
(general).
§ 73. The Logic of Science is the Syntax
OF THE Language of Science
In what follows we shall examine the nature of the questions of
the logic of science in the wide sense, including, as already indi-
cated, the so-called philosophical problems concerning the founda-
tions of the individual sciences, and we shall show that these
questions are questions of syntax. In order to do this, it must first
be shown that the object-questions which occur in the logic of
science (for example, questions concerning numbers, things, time
and space, the relations between the psychical and the physical,
etc.) are only pseudo-object-questions — i.e. questions which, be-
cause of a misleading formulation, appear to refer to objects while
actually they refer to sentences, terms, theories, and the like — and
are, accordingly, in reality, logical questions. And secondly, it
must be shown that all logical questions are capable of formal
presentation, and can, consequently, be formulated as syntactical
questions. According to the usual view, all logical investigation
comprises two parts: a formal inquiry which is concerned only
with the order and syntactical kind of the linguistic expressions,
and an inquiry of a material character, which has to do not merely
with the formal design but, over and above that, with questions of
meaning and sense. Thus the general opinion is that the formal
problems constitute, at the most, only a small section of the domain
282
PART V. PHILOSOPHY AND SYNTAX
of logical problems. As opposed to this, our discussion of general
syntax has already shown that the formal method, if carried far
enough, embraces all logical problems, even the so-called pro-
blems of content or sense (in so far as these are genuinely logical
and not psychological in character). Accordingly, when we say
that the logic of science is nothing more than the syntax of the
language of science, we do not mean to suggest that only a certain
number of the problems of what has hitherto been called the logic
of science (as they appear, for example, in the works previously
mentioned) should be regarded as true problems of the logic of
science. The view we intend to advance here is rather that all
problems of the current logic of science, as soon as they are
exactly formulated, are seen to be syntactical problems.
• Jt was Wittgenstein who first exhibited the close connection between
the logic of science (or ‘philosophy**, as he calls it) and syntax. In
particular, he made clear the fonnal nature of logic and emphasized
the fact that the rules and proofs of syntax should have no reference
to the meaning of symbols ([Tractatus^, pp. 52, 56, and 164).
Further, he has shown that the so-called sentences of metaphysics
and of ethics are pseudo-sentences. According to him philosophy is
“critique of language** (op. cit. p. 62), its business is “the logical
clarification of ideas’* (p. 76), of the sentences and concepts of
science (natural science), that is, in our terminology, the logic of
science. Wittgenstein *s view is represented, and has been further
developed, by the Vienna Circle, and in this part of the book I owe
a great deal to his ideas. If I am right, the position here maintained
is in general agreerpent with his, but goes beyond it in certain im-
portant respects. In what follows my view will sometimes be con-
trasted with his, but this is done only for the sake of greater clarity,
and our agreement on important fundamental questions must not
therefore be overlooked. -
There are two points especially on which the view here presented
differs from that of Wittgenstein, and specifically from his negative
theses. The first of these theses (op. cit. p. 78) states: “Propositions
cannot represent the logical form : this mirrors itself in the proposi-
tions. That which mirrors itself in language, language cannot repre-
sent. That which expresses itself in language, we cannot express by
language If two propositions contradict one another, this is
shown by their structure ; similarly, if one follows from another, etc.
What can be shown cannot be said. ... It would be as senseless to
ascribe a formal property to a proposition as to deny it the formal
property.** In other words : There are no sentences about the forms
of sentences ; there is no expressible syntax. In opposition to this view,
our construction of syntax has shown that it can be correctly formu-
lated and that syntactical sentences do exist. It is just as possible to
§73- logic of science the syntax of language of science 283
construct sentences about the forms of linguistic expressions, and
therefore about sentences, as it is to construct sentences about the
geometrical forms of geometrical structures. In the first place, there
are the analytic sentences of pure syntax, which can be applied to the
forms and relations of form of linguistic expressions (analogous to
the analytic sentences /)f arithmetical geometry, which can be ap-
plied to the relations of form of the abstract geometrical structures) ;
and in the second place, the synthetic physical sentences of de-
scriptive syntax, which are concerned with the forms of the linguistic
expressions as physical structures (analogous to the synthetic em-
pirical sentences of physical geometry, se^ § 25). Thus syntax is
exactly formulable in the same way as geometry is.
Wittgenstein’s second negative thesis states that the logic of
science (‘‘ philosophy ”) cannot be formulated. (For him, this thesis
does not coincide with the first, since he does not consider the logic
of science and syntax to be identical ; see below.) “ Philosophy is not
a theory, but an activity. A philosophical work consists essentially
of elucidations. The result of philosophy is not a number of * philo-
sophical propositions,* but to make propositions clear” (p. 76).
Consistently Wittgenstein applies this view to his own jvork*also;
at the end he says : ” My propositions are elucidatory in this way : he
who understands me finally recognizes them as senseless, when he
has climbed out through them, on them, over them. (He must, so to
speak, throw away the ladder, after he has climbed up on it.) He
must surmount these propositions ; then he sees the world rightly.
Whereof one cannot speak, thereof one must be silent” (p. 188).
According to this, the investigations of the logic of science contain
no sentences, but merely more or less vague explanations which the
reader must subsequently recognize as pseudo-sentences and
abandon. Such an interpretation of the logic of science is certainly
very unsatisfactory. [Ramsey first raised objections to Wittgen-
stein’s conception of philosophy as nonsense, but important non-
sense {{Foundations}, p. 263), and then Neurath, in particular,
{{Soziol. Phys.}, pp. 395 f. and {Psychol.}, p. 29) definitely rejected it.]
When in what follows it is shown that the logic of science is syntax,
it is at the same time shown that the logic of science can be formu-
lated, and formulated not in senseless, if practically indispensable,
pseudo-sentences, but in perfectly correct sentences. The diflFerence
of opinion here indicated is not merely theoretical; it has an im-
portant influence on the practical form of philosophical investiga-
tions. Wittgenstein considers that the only difference between the
sentences of the speculative metaphysician and those of his own and
other researches into the logic of science is that the sentences of the
logic of science — which he calls philosophical elucidations — in spite
of their theoretical lack of sense, exert, practically, an important
psychological influence upon the philosophical investigator, which
the properly metaphysical sentences do not, or, at least, not in the
same way. Thus there is only a difference of degree, and that a very
PART V. PHILOSOPHY AND SYNTAX
284
vague one. The fact that Wittgenstein does not believe in the possi-
bility of the exact formulation of the sentences of the logic of science
has as its consequence that he does not demand any scientific
exactitude in his own formulations, and that he draws no sharp line
of demarcation between the formulations of the logic of science and
those of metaphysics. In the following discussion we shall see that
translatability into the formal mode of speech — that is, into syn-
tactical sentences — is the criterion which separates the proper sen-
tences of the logic of science from the other philosophical sentences
— ^we may call them metaphysical. In some of his formulations,
Wittgenstein has clearly overstepped this boundary; this conse-
quence of his belief in the two negative theses is psychologicallv
quite understandable.
In spite of this difference of opinion, I agree with Wittgenstein
that there are no special sentences of the logic of science (or philo-
sophy). The sentences of the logic of science are formulated as
syntactical sentences about the language of science; but no new
dolnain in addition to that of science itself is thereby created. The
sentences of syntax are in part sentences of arithmetic, and in part
sentences,, of physics, and they are only called syntactical because
they are concerned with linguistic constructions, or, more speci-
fically, with their formal structure. Syntax, pure and descriptive, is
nothing more than the mathematics and physics of language.
Wittgenstein says of the rules of logical syntax (see above) that
they must be formulated without any reference to sense or meaning.
According to our view the same thing holds also for the sentences
of the logic of science. But Wittgenstein, as it appears, thinks that
these sentences (the so-called philosophical elucidations) go beyond
the formal and refer to the sense of the sentences and terms. Schlick
{[Wende\ p. 8) interprets Wittgenstein's position as follows : philo-
sophy “ is that activity by which the meaning of propositions is estab-
lished or discovered"; it is a question of “what the propositions
actually mean. The content, soul, and spirit of science naturally con-
sist in what is ultimately meant by its sentences ; the philosophical
activity of rendering significant is thus the alpha and omega of all
scientific knowledge".
§ 74. Pseudo-Object-Sentences
We have already distinguished (in an inexact manner) between
object-sentences and logical sentences. We will now contrast in-
stead (at first also in an inexact manner) the two domains of object-
sentences and syntactical sentences^ only those logical sentences
which are concerned with form being here taken into account and
included in the second domain. Now there is an intermediate field
between these two domains. To this intermediate field we will
§ 74 * PSEUDO-OBJECT-SKNTENCES 285
assign the sentences which are formulated as though they refer
(either partially or exclusively) to objects, while in reality they
refer to syntactical forms, and, specifically, to the forms of the
designations of those objects with which they appear to deal. Thus
these sentences are syntactical sentences in virtue of their con-
tent, though they are clisguised as object-sentences. We will call
them pseudo-object-sentences. If we attempt to represent in a
formal way the distinction which is here informally and inexactly
indicated, we shall see that these pseudo-object-sentences are
simply quasi^syntactical sentences of the material mode of speech
(in the sense already formally defined, see § 64).
To this middle territory belong many of the questions and sen-
tences relating to the investigation of what are called philosophical
foundations. We will take a simple example. Let us suppose that
in a philosophical discussion about the concept of number we
want to point out that there is an essential difference between
numbers and (physical) things, and thereby to give a* warning
against pseudo-questions concerning the place, weight, and so on
of numbers. Such a warning will probably be formulated as a
sentence of, say, the following kind: “Five is not a thing but a
number” (Si). Apparently this sentence expresses a property of
the number five, like the sentence “ Five is not an even but an odd
number” (S2). In reality, however. Si is not concerned with the
number five, but with the word ‘ five ’ ; this is shown by the formu-
lation Sa which is equipollent to Sii “‘Five* is not a thmg-word
but a number-word.** While S2 is a proper object-sentence, Si is
a pseudo-object-sentence; Si is a quasi-syntactical sentence
(material mode of speech), and S3 is the correlated syntactical
sentence (formal mode of speech).
We have here left out of account those logical sentences which
assert something about the meanings content^ or sense of sentences
or linguistic expressions of any domain. These also are pseudo-
object-sentences. Let us consider as an example the following
sentence. Si*. “Yesterday’s lecture was about Babylon.” Si ap-
pears to assert something about Babylon, since the naiiie ‘ Babylon *
occurs in it. In reality, however, Si says nothing about the town
Babylon, but merely something about yesterday’s lecture and the
word ‘ Babylon ’. This is easily shown by the following non-formal
286
PART V. PHILOSOPHY AND SYNTAX
Babylon it does not matter whether Si is true or false. Further,
that Si is only a pseudo-object-sentence is clear from the circum-
stance that Si can be translated into the following sentence of
(descriptive) syntax: “In yesterday’s lecture either the word
‘ Babylon * or an expression synonymous with the word ‘ Babylon *
occurred” (S2)*
Accordingly, we distinguish three kinds of sentences:
OhjecUsentences 2. Pseudo-object- 3. Syntactical
sentences = quasi- sentences
syntactical sentences
Material mode of Formal mode of
speech speech
Examples: “5 is Examples Five is Examples ‘ Five ’
prime number”; not a thing, but a is not a thing- word,
“Babylon was a big number”; “Babylon but a number-word”;
town”; “lions are was treated of in “ the word ‘ Babylon *
mammals.” yesterday’s lecture.” occurred in yes-
(“Five is a number- terday’s lecture”;
word” is an example ‘“A. is a con-
belonging to the au- tradictory sentence. ”
tonymous mode of
speech.)
The intermediate field of the pseudo-object-sentences, the
boundaries of which have so far been only materially and inexactly
indicated, can also be exactly, and moreover formally, demarcated.
The pseudo-object-sentences are, namely, quasi-syntactical sen-
tences of the material mode of speech. [We can leave the autony-
mous mode of speech out of account here, since there is practically
no danger of a sentence belonging to this mode of speech being
mistaken for an object-sentence.] The criterion of the material
mode of speech assumes a simpler form when we are concerned
with an object-language which contains its own syntax-
language S2 as a sub-language. For instance, let be the English
language representing the whole language of science ; then the
syntax-language Sg, in which the syntax of is formulated, is a
sub-language of Sj. This expresses the fact that we regard syntax
not as a special domain outside that of the rest of science but as a
sub-domain of science as a whole, which forms a single system
(Neurath: Einheitswissenschaft) having a single language Sj.
That a language may contain its own syntax without contradiction
§ 74 - PSEUDO-OBJECT-SENTENCES 287
we have already shown. Even if the syntax-language Sg is a sub-
language of Sj it is, of course, both possible and necessary to dis-
tinguish between a sentence S^, of Sj (which may also belong to
Sg), and a syntactical sentence 0g, concerning Si, which belongs to
Sg and therefore also to Sj. For simplicity’s sake, we will formulate
the criterion of the material mode of speech for the simplest sen-
tential form only (and further, for the sake of brevity and clarity,
we will formulate it for a symbolic sentence) (see § 64). Let Si be
‘ P (a) ’ ; Si is called quasi-syntactical in respect of ‘ a *, if there exists
a syntactical predicate ‘Q* such that ‘P(a)* is equipollent to
‘Q(‘a’)’ (Sg) and ‘P(b)’ is equipollent to ‘Q(‘b*)’, and corre-
spondingly for every expression isogenous with ‘ a Now ‘ P * may
possibly be a syntactical predicate which is equivalent in meaning
to ‘Q* (this would be shown formally by the fact that ‘P(‘a*)*
would also be a sentence, and moreover a sentence equipollenf to
‘Q(‘a*)*, and that, further, ‘P(‘b*)* would be equipollent to
‘Q(‘b*)’, and correspondingly for every expression hogenous
with ‘a’); if this is not the case, we call Si a sentence of the
material mode of speech. ‘ Q ’ is called a syntactical predicate corre-
lated to the quasi-syntactical predicate ‘ P ’ ; and S2 is called a syn-
tactical sentence correlated to the quasi-syntactical sentence Sj.
In the translation from the material to the formal mode of speech,
Si is translated into Sg.
In order to make it clearer and facilitate its practical application
to the following examples, we will formulate the criterion (still for
the simplest form of sentence) once more, in a less exact, non-formal
way (the examples of sentences which come later, especially those
of the logic of science, belong almost entirely to the word-language ;
in consequence, they are themselves not formulated sufficiently
exactly to make possible the application to them of exact concepts).
Si is called a sentence of the material mode of speech if Si asserts
a property of an object which has, so to speak, parallel to it,
another, and syntactical, property; that is to say, when there is a
syntactical property which belongs to a designation of an object
if, and only if, the original property belongs to the object.
It is easy to see that in the previous example concerning * Baby-
lon * this criterioir is fulfilled for the sentence Si : the syntactical
(in this case the descriptive-syntactical) property which is asserted
in Sg of the word ‘ Babylon ’ is parallel to that property which is
288
PART V. PHILOSOPHY AND SYNTAX
asserted in Si of the town of Babylon ; for if, and only if, yesterday's
lecture was concerned with a certain object, did a designation of
that object occur in the lecture. The criterion of the material mode
of speech is likewise fulfilled for the sentence Si of the example
concerning ‘five*; for if, and only if, the property expressed
in Si — that of being not a thing but a number — belongs to some
object (for instance, to the number five) does the property ex-
pressed in S2 — that of being not a thing-word but a number-
word — belong to a designation of this object (in the example, to
the word ‘ five *).
§ 75. Sentences about Meaning
In this section, we shall consider various kinds of sentences of
the' material mode of speech, especially those kinds which occur
frequently in philosophical discussions. On the basis of these in-
vestigations we shall be better able to diagnose the material mode
of speech in subsequent cases. Further, by this means the whole
character of philosophical problems will become clearer to us. The
obscurity with regard to this character is chiefly due to the de-
ception and self-deception induced by the application of the
material mode of speech. The disguise of the material mode of
speech conceals the fact that the so-called problems of philo-
sophical foundations are nothing more than questions of the logic
of science concerning the sentences and sentential connections of
the language of science, and also the further fact that the questions
of the logic of science are formal — that is to say, syntactical —
questions. The true situation is revealed by the translation of the
sentences of the material mode of speech, which are really quasi-
syntactical sentences, into the correlated syntactical sentences and
thus into the formal mode. We do not mean by this that the
material mode of speech should be entirely eliminated. Since it is
in general use and often easier to understand, it may well be re-
tained in its place. But it is a good thing to be conscious of its use,
so as to avoid the obscurities and pseudo-problems which other-
wise easily result from it.
In a sentence Si of the material mode of speech, the illusion that
a genuine object-sentence is present is most easily dissipated if Si
belongs in part to^the syntax-language Sg, but contains at the same
§ 75 - SENTENCES ABOUT MEANING 289
time elements of Sj which do not belong to Sg. [Not all sentences
of this kind are sentences of the material mode of speech. For
example, the sentence “The University of Freiburg bears the
inscription ‘the truth will make you free*** is not a quasi-syn-
tactical sentence but a simple sentence of descriptive syntax.]
Especially important here are those sentences which express a re-
lation of designation, that is to say, those in which one of the
following expressions occurs; ‘treats of*, ‘speaks about*, ‘means’,
‘signifies*, ‘names*, ‘is a name for’, ‘designates’, and the like.
We shall now give a series of such sentences concerning meanings,
and, along with them, the correlated syntactical sentences. The
first of these examples has already been discussed. [It is, of course,
of no importance whether or not the sentences in the examples are
true. ] ^ .
Material mode of speech
(quasi-syn tactical
sentences)
1 a. Yesterday’s lecture treated
of Babylon.
2 a. The word ‘ daystar ’ desig-
nates (or; means', or: is a name
for) the sun.
3 a. The sentence Sj means
(or: asserts; or: has the content;
or: has the meaning) that the
moon is spherical.
4 a. The word ‘luna’ in the
Latin language designates the
moon.
5 a. The sentence ‘ ’ of the
Chinese language means that the
moon is spherical.
Formal mode of speech
(the correlated syntactical
sentences)
I h. In yesterday’s lecture the
word ‘Babylon’ (or a synony-
mous designation) occurred.
zb. The word ‘daystar’ is
synonymous with ‘ sun ’.
36. ©1 is equipollent to the
sentence ‘The moon is spheri-
cal.’
4^. There is an equipollent
expressional translation of the
Latin into the English language
in which the word ‘ moon ’ is the
correlate of the word ‘ luna ’.
56. There is an equipollent
sentential translation of the
Chinese into the English lan-
guage in which the sentence
‘The moon is spherical’ is the
correlate of the sentence ‘ . . . *.
The following examples, 6 and 7, show how the difference be-
tween the meaning of an expression and the object designated by the
expression can be formally represented. [This difference is em-
phasized by the phenomenologists, but explained only in a psycho-
logical, not in the logical, sense.]
290
PART V. PHILOSOPHY AND SYNTAX
6a. The expressions ‘merle* 6 b. ‘Merle* and ‘blackbird*
and ‘ blackbird * have the same are L-synonymous.
meaning (or : mean the same ; or :
have the same intensional object),
7 a. ‘Evening star* and ‘mom- 76. ‘Evening star* and
ing star 'have a different meaning, ‘morning star’ are not L-syn^
buttheydejigwafe the same object, onymous, but P-synonymous.
[With respect to a symbolic (P-) language, the above correlates
may also be formulated thus :6 b. ‘ = 9I2 * is analytic. 7 b. ‘ = ^4 *
is not analytic but P-valid.]
In the case of sentences the formal representation of the difference
between the fact designated and the meaning is analogous. [The usual
formulations like ‘mean the same* or ‘have the same content* are
ambiguous ; in some cases 8 6 is intended, in others 9 6, and in many
the intention remains obscure.]
8 a. The sentences and <^2 86. and S2 are L-equi-
havo the same meaning. pollent.
9 a. Si and 02 have a dif- 96. Si and S2 are not L-
ferent meaning but they represent equipollent but P-equipollent.
(or: describe) the same fact.
[With respect to a symbolic language: 8 6. ‘ Si= Sg’ is analytic.
96. ‘ Si = 02* is not analytic but P-valid.]
10 a. The sentences of arith- I 106. The sentences of arith-
metic state (or: express) certain metic are composed of numerical
properties of numbers and cer- expressions and one- or many-
tain relations bemeen numbers, termed numerical predicates
combined in such and such a way.
1 1 a. A particular sentence of 1 1 6. A particular sentence of
physics states the condition of a physics consists of a descriptive
spatial point at a given time, predicate and spatio-temporal
co-ordinates as arguments.
The following examples 12 a, 13 a, and 14 a appear at first to be of
the same kind as i a and 4 a. Actually, however, they demonstrate
particularly clearly the danger of error which is involved in the use
of the material mode of speech.
12 a. This letter is about the 126. In this letter a sentence
son of Mr. Miller. occurs in which ^1 is the
description ‘ the son of Mr.
Miller*.
The expression ‘le che- 136. There is an equipollent
val de M * designates (or : means) expressional translation from the
the horse of M. French into the English lan-
guage in which ‘ the horse of M ’ is
the correlate of ‘ le che val de M *.
14 a. The expression ‘ un ^ 14 - 146. (Analogous to 1 3 6.)
phant bleu* means a blue ele-
phant.
291
§ 75 - SENTENCES ABOUT MEANING
Let us assume that Mr. Miller has no son ; even in this caoe the
sentence 12a may still be true; the letter will then merely be telling
a lie. Now, from the true sentence iza, according to the ordinary
logical rules of inference, a false sentence can be derived. In order
to make the derivation more exact, we will use a symbolism in place
of the word-language. Instead of ‘ this letter ’ we will write ‘ b * ; in-
stead of ‘ b is about a ’ we will write ‘ H (b, a) ’ ; and instead of ‘ the son
of a’ we will write ‘Son"a* (descriptional in Russell’s symbolism,
see § 38^:). Hence for 12^2 will be written: ‘ H (b, Son' Miller) ’
( 0 i). According to a well-known theorem of logistics (see my
[Logisttk\y § 7 c : L 7.2), from a sentence Cilrn) in which a descrip-
tion occurs as argument, a sentence is derivable which asserts that
there exists something which has the descriptional property.
Accordingly, from would be derivable ‘ (3 (Son (x, Miller)) ’
(S2) ; or, in words : “ a son of Mr. Miller exists ”. This, however, is a
false sentence. Similarly the possibly false sentence “There is a
horse of M“ is derivable from 13a, and the false sentence “there is
a blue elephant” from 14a. On the other hand, by the usual rifle's
no false sentences can be derived from the sentences 126, 136, and
146 of the formal mode of speech. These examples show that the
use of the material mode of speech leads to contradictions if the
methods of inference which are correct for other sentences are
thoughtlessly used also in connection with it. [It cannot be main-
tained that the formulations 12 a, 13 a, and 14 a are incorrect, or that
the use of the material mode of speech leads necessarily to contra-
dictions ; for, after all, the word-language is not bound by the rules
of logistics. If, therefore, one wishes to admit the material mode of
speech, one must apply to it a system of rules which is not only more
complicated than that of logistics but is also more complicated than
that which governs the rest of the sentences of the word-language.]
Some sentences contain a relation of meaning which is to some
extent concealed. With sentences of this kind it is not obvious, at
first sight, that they belong to the material mode of speech. The
most important examples of this are the sentences which use the
so-called indirect or oblique mode of speech (that is to say, sen-
tences which say something about a spoken, thought, or written
sentence, but which do so not by a statement of the original word-
ing but instead by means of a ‘that*, ‘whether’, or other ‘w. . .*
sentence, or of a subordinate sentence without a connective word,
or of an infinitive with ‘to’). In the following examples, 15 a and
16 a, the formulations 15^ and 16^ show that the sentences in
which the indirect mode of speech occurs are of the same kind as
the examples previously discussed, and hence also belong to the
material mode of speech.
292
PART V. PHILOSOPHY AND SYNTAX
I. Material mode of speech
I. Sentences in in-
direct speech
15 a. Charles said
(wrote, thought) Peter
was coming tomorrow
(or: that Peter was
coming tomorrow).
1 6 a. Charles said
where Peter is.
2. Sentences about
meaning
i$h, Charles said
a sentence which
means that Peter is
coming tomorrow.
166. Charles said
a sentence which
states where Peter is.
II. Formal mode
of speech
15 c. Charles said
the sentence ‘ Peter is
coming tomorrow *(or :
a sentence of which
this is a consequence).
i6r. Charles said
a sentence of the
form ‘Peter is — ’in
which a spatial desig-
nation takes the place
of the dash.
The use of the indirect mode of speech is admittedly short and
convenient ; but it contains the same dangers as the other sentences
of the material mode. For instance, sentence 15^1, as contrasted
with sentence 1 5 c, gives the false impression that it is concerned
with Peter, while in reality it is only concerned with Charles and
with the word ‘ Peter When the direct mode of speech is used,
this danger does not occur. For instance, the sentence: “Charles
says ‘ Peter is coming tomorrow*’’ does not belong to the material
mode of speech : it is a sentence of descriptive syntax. The direct
mode of speech is the ordinary form used in the word -language for
the formal syntactical mode. (On the construction of the syn-
tactical designation of an expression with the help of inverted
commas, see § 41.)
The examples so far given suffice to show that, with certain
formulations in the material mode of speech, there is the danger of
obscurity or of contradictions. It is true that in such simple cases
as these the danger is easy to avoid. But in less obvious cases of
essentially the same kind, especially in philosophy, the application
of the material mode of speech has time and again led to incon-
sistencies and confusions.
§ 76. Universal Words
We will call a predicate of which every full sentence is an ana-
lytic sentence a universal predicate^ or, if it is a word in the word-
language, a universal word. [For every genus of predicates a uni-
§ 76 . UNIVERSAL WORDS 293
versal predicate can easily be defined. For instance, if ptj is a pr^
of any genus whatsoever, we define the universal predicate pr2, of
the same genus, as follows: pr2(Di)= (pri(Di)V ^pri(oi)).] The
investigation of universal words is especially important for the
analysis of philosophical sentences. They occur very often in such
sentences both in metaphysics and in the logic of science, and are
for the most part in the material mode of speech. In order to
facilitate the practical application of the criterion for ‘universal
word let us also formulate it in an informal way. A word is called
a universal word if it expresses a property (or relation) which be-
longs analytically to all the objects of a genus, any two objects
being assigned to the same genus if their designations belong to the
same syntactical genus. Since the rules of syntax of the word-
language are not exactly established, and since linguistic usage
varies considerably on just this point of the generic classification
of words, our examples of universal words must always be given
with the reservation that they are valid only for one particular use
of language.
Examples: i. ‘Thing* is a universal word (provided that the desig-
nations of things constitute a genus). In the word-series ‘dog*,
‘ animal ’, ‘ living creature *, ‘ thing *, every word is a more compre-
hensive predicate than the previous one, but only the last is a uni-
versal predicate. In the corresponding series of sentences, ‘ Caro is
a dog *, ‘ . . . is an animal *, ‘ . . , a living creature *, ‘ Caro is a thing *, the
content is successively diminished. But the final sentence is funda-
mentally different from the preceding ones, in that its L-content is
null and it is analytic. If in ‘ Caro is a thing *, ‘ Caro * is replaced by
any other thing-designation, the result is again an analytic sentence ;
but if ‘Caro* is replaced by an expression which is not a thing-
designation, the result is not a sentence at all.
2. ‘Number’ is a universal word (provided that the numerical
expressions constitute a genus, as for instance in Languages I and
II, as opposed to Russell’s language where they form a part of the
class-expressions of the second level). In the series of predicates,
‘ number of the form 2" -f i *, ‘ odd number *, ‘ number ’, only the last
is a universal predicate. In the series of sentences ‘ 7 has the form
2”+ I *, ‘7 is odd*, ‘7 is a number*, the second is already analytic,
but only the third has the property that every sentence which re-
sults from it if * 7 * is replaced by another 3 is again analytic. If ‘ 7 ’
is replaced by an expression which is not a 3. then no sentence re-
sults (on the assumptions made at the beginning).
Examples of universal words: ‘thing’, ‘object’, ‘property’,
‘ relation ’, ‘ fact ’, ‘ condition ’, ‘ process *, ‘ event ’, ‘ action ’, ‘ spatial
294 part V. PHILOSOPHY AND SYNTAX
point*, ‘spatial relation*, ‘space* (system of spatial points con-
nected by spatial relations), ‘ temporal point *, ‘ temporal relation *,
‘time* (system of temporal points connected by temporal rela-
tions); ‘number’, ‘integer* (in I and II), ‘real number* (in some
systems), ‘function’, ‘aggregate* (or ‘clasj^*); ‘expression* (in a
language of pure syntax) ; and many others.
We all use such universal words in our writings in almost every
sentence, especially in the logic of science. That the use of these
words is necessary is, however, only due to the deficiencies of the
word-languages, i.e. to their inadequate syntactical structure.
Every language can be transformed in such a way that universal
words no longer occur in it, and this without any sacrifice either of
expressiveness or conciseness.
We will now distinguish two methods of employing universal
words (without making an exact and formal differentiation). The
second method involves the material mode of speech, and will be
dealt with later. The first method has to do with genuine object-
sentences. Here a universal word serves to point out the syn-
tactical genus of another expression. In some cases the syntactical
genus of the other expression is already univocally determined by
its form alone ; the special indication of it by means of the added
universal word is then only of use in making it more prominent,
as an aid to the comprehension of the reader. In other cases, how-
ever, the addition of the universal word is necessary, since without
it the other expression would be ambiguous. In all these cases of
the first way of using it, the universal word is, so to speak, de-
pendent-, it is an auxiliary grammatical symbol added to another
expression, something like an index.
Examples: i. “By means of the process of crystallization. .
Since crystallization belongs without any ambiguity to the genus of
the processes, one might simply say : “ By means of crystallization. ...”
Here the universal word ‘ process ’ only serves to. point out the genus
to which the word ‘crystallization* belongs. Similarly in the fol-
lowing examples: 2. “The condition of fatigue....’’ 3. “The num-
ber five....”
In the following sentences the universal word is necessary for
univocality. It can be rendered superfluous by the use of a suffix
(‘ 7 * and ‘ 7r ’) or by introducing various explicit expressions in place
of the ambiguous one. 4a. “The integer 7....” 4 6. “ The real num-
ber 7....” 5<i. “ The condition of friendship....” 56. “The relation
of friendship....**
§76. UNIVERSAL WORDS 295
In the word -language universal words are especially needed as
auxiliary symbols for variables^ that is, in the formulation of uni-
versal and existential sentences, for the purpose of showing from
which genus the substitution- values are to be taken. The word-
language employs as variables words (‘ a *, ‘ some ‘ every *, ‘ all *,
* any *, and so on) to which no particular genus is correlated as their
realm of values. If, as is usual in the symbolic languages, different
kinds of variables were used for the different genera of substitution-
values, the addition of a universal word would be superfluous.
Accordingly, the universal word here serves to some extent as an
index to a variable, which indicates the genus of its substitution-
values.
Examples: We will contrast the formulations of the word-language
with those of the symbolic language of logistics. 6 a. “If any num-
ber..., then....” 66. “(jc) (... D ...)” (where ‘ a:’ is a 3). 7a. “There
is a number....” 7 6. “ (3 jc) (...)” (where ‘x* is a 3). 8a. “I know a
thing which....” 86. ” (3 (where * x" is a thing-variable).
9a. “Every numerical property....” 96. “(F) (...)” (where ‘F* is
a p of which the values are 3pr^). 10 a. “There is a relation....”
106. “(3F) (...)” (where ‘F* is a p^).
Wittgenstein [ Tractatus] p. 84 says : “ So the variable name ‘ jc * is the
proper sign of the pseudo-concept object. Wherever the word * ob-
ject* (‘thing’, ‘entity*, etc.) is rightly used, it is expressed in logical
symbolism by the variable name Wherever it is used otherwise,
i.e. as a proper concept-word, there arise senseless pseudo-proposi-
tions. . . . The same holds of the words ‘ complex *, ‘ fact *, ‘ function *,
‘ number *, etc. They all signify formal concepts and are presented in
logical symbolism by variables, not by functions or classes (as Frege
and Russell thought). Expressions like ‘ i is a number *, ‘ there is only
one number nought*, and all like them are senseless.*’ Here the
correct view is taken that the universal words designate formal (in
our terminology: syntactical) concepts (or, more exactly: are not
syntactical but quasi-syntactical predicates) and that in the transla-
tion into a symbolic language they are translated into variables (or,
again more exactly : they determine the kind of variables by which
the words ‘ a *, ‘ every *, and so on, are translated ; it is only the kind
of variables that is determined, and not their design; in the
examples given above, or * z* can equally well be taken instead
of ‘jc*). On the other hand, I do not share Wittgenstein’s opinion
that this method of employing the universal words is the only ad-
missible one. We shall see later that, precisely in the most important
cases, there is another method of use in which the universal word is
employed independently (“ as a proper concept-word **). There it is
a question of sentences of the material mode of speech which are to
be translated into syntactical sentences. Sentences of this kind with
296 PART V. PHILOSOPHY AND SYNTAX
a universal word are held by Wittgenstein to be nonsense, because
he does not consider the correct formulation of syntactical sentences
to be possible.
The use of universal words in questions in connection with one of
the w... interrogatives (‘what*, ‘who*, ‘where*, ‘which*, etc.)
is akin to their use in universal and existential sentences. Here
also, in translation into a symbolic language, the universal word
determines the choice of the kind of variable. A yes-or-no ques-
tion demands either the affirmation or the denial of a certain sen-
tence Sj, that is to say, the assertion of either Si or
[Example: The question “Is the table* round?** requires us to
assert in answer either: “the table is round** or: “the table is not
round.’*] As contrasted with this, aw... question demands in
reference to* a certain sentential function the assertion of a closed
full sentence (or sentential framework). In a symbolic question,
the genus of the arguments requested is determined by the kind of
the argument variables. In the word-languages this genus is in-
dicated by means either of a specific w. . . interrogative (such as
‘ who *, ‘ where *, ‘ when *) or of an unspecific w . . . interrogative
(such as ‘ what *, ‘ which *) with an auxiliary universal word. Hence
here also the universal word is, so to speak, an index to a variable.
Examples: i. Suppose I want to ask someone to make an assertion
of the form “ Charles was — in Berlin **, where a time-determination
of which I am ignorant but which I wish to learn from the assertion
is to take the place of the dash. Now the question must indicate by
some means that the missing expression is to be a time-determina-
tion. If symbols are used this can be effected by giving a sentential
function in which in the place of the argument a variable ‘ t *, which
is established as a temporal variable, occurs. [To symbolize the
question, the variable whose argument is requested must be bound
by means of a question-operator, e.g. ‘ ( ? /) (Charles was t in Berlin) *. ]
In the word-language the kind of argument requested is made known
either by means of the specific question-word ‘when* (“When was
Charles in Berlin ? **) or by means of the universal word ‘ time * or
‘temporal point’ attached to an unspecific question-word (“At
what time was Charles in Berlin?’*).
2. I wish to ask someone to make me an assertion of the form
“ Charles is — of Peter’’, where a relation-word is to take the place
of the dash (‘father*, ‘friend’, ‘teacher’, or the like). The symbolic
formulation of this question, by means of the relational variable ‘i? ’,
is : ‘ ( ? /?) (i? (Charles, Peter)) ’. Its formulation in the word-language
by means of the addition of the universal word ‘relation* to an
unspecific question-word is: “What relation is there between
Charles and Peter?**
§ 77 * universal words in material mode of speech 297
§ 77. Universal Words in the Material
Mode of Speech
In the first use of the universal word, which we have up to now
been discussing, it appears as an auxiliary symbol determining the
genus of another expression ; it was found that, if in place of this
other expression a symbol indicating its own genus was introduced,
then the universal word could be dispensed with. As opposed
to this, in the second use the universal word appears as an inde-
pendent expression, which in the simplest form occupies the place
of the predicate in the sentence in question. Sentences of this kind
belong to the material mode of speech ; for a universal word is here
a quasi -syntactical predicate; the correlated syntactical predicate
is that which designates the appertaining expressional gequs.
\Example : ‘ number * is a universal word because it belongs ana-
lytically to all the objects of a genus of objects, namely, that of the
numbers; the correlated syntactical predicate is ‘numerical ex-
pression * (or ‘ number-word *), since this applies to all expressions
which designate a number. The sentence “ Five is a number** is a
quasi-syntactical sentence of the material mode of speech ; a corre-
lated syntactical sentence is ‘“Five* is a number- word **. ]
Sentences with universal Syntactical sentences
words
(Material mode of speech) (Formal mode of speech)
17 a. The moon is a thing ; five 176. ‘ Moon * is a thing-word
is not a thing, but a number, (thing-name) ; ‘ five * is not a
thing-word, but a number-word.
In 17a, as contrasted with sentences like “the thing moon...**,
“ the number five. . .**, the universal words ‘ thing * and ‘ number * are
independent.
1 8 a. A property is not a thing, iHh. An adjective (property-
word) is not a thing-word.
That the formulation 18 a is open to objection is shown by the
following consideration. 18 a violates the ordinary rule of types.
This comes out particularly clearly when an attempt is made to
formulate it symbolically, either by means of ‘(F) (Prop (F) 3
Thing (F))* or by means of ‘ (x) (Prop (:x:) D Thing (a;))*; in the
first case, ‘ Thing (F) *, and in the second case ‘ Prop (x) *, is incon-
sistent with the rule of types. Therefore, if 18 a is admitted as a
sentence (it makes no difference whether true or false), by the usual
syntax of logistics RusselPs antinomy can be constructed. If this is
to be avoided, special complicated syntactical rules are necessary.
PART V. PHILOSOPHY AND SYNTAX
298
19a. Friendship is a re/a 196. ‘Friendship* is a rela-
tion-word.
20a. Friendship is not a pro- 206. ‘Friendship* is not a
perty, property- word.
19 a corresponds to the sentential form used by Russell
‘ cRel*; the analogous symbolic formulation of 20a would, how-
ever, violate the rule of types. On the other hand, the correlated
sentences of the formal mode of speech, 196 and 206, are, even
without any special preliminary adjustments, of the same kind and
equally correct. In contrast with the pseudo-object-sentence 19 a,
a sentence of the form “Friendship ensues if...*’, for instance, is a
genuine object-sentence, and therefore not a sentence of the material
mode of speech.
It is frequently said that the rule of types (even the simple one)
restricts the expressiveness of a language to an inconvenient extent,
and that one is often tempted to use formulations which would not
be allowed by it. Such formulations, however, are often (like the
examples given) only pseudo-object-sentences with universal
words. If, in such cases, instead of the object-terms which one
would like to, but must not, combine, one uses the correlated syn-
tactical terms, the restrictive effect of the rule of types disappears.
Independent universal words appear very often in philosophical
sentences, in the logic of science as well as in traditional philo-
sophy, Most of the examples of philosophical sentences which will
be given later belong to the material mode of speech by reason of the
employment of independent universal words.
§ 78. Confusion in Philosophy Caused by the
Material Mode of Speech
The fact that, in philosophical writings — even in those which are
free from metaphysics — obscurities so frequently arise, and that in
philosophical discussions people so often find themselves talking
at cross purposes, is in large part due to the use of the material
instead of the formal mode of speech. The habit of formulating in
the material mode of speech causes us, in the first place, to deceive
ourselves about the objects of our own investigations: pseudo-
object-sentences mislead us into thinking that we are dealing with
extra-linguistic objects such as numbers, things, properties, ex-
periences, states of affairs, space, time, and so on ; and the fact that,
in reality, it is 'a case of language and its connections (such as
§ 78 . CONFUSION CAUSED BY THE MATERIAL MODE 299
numerical expressions, thing-designations, spatial co-ordinates,
etc.) is disguised from us by the material mode of speech. This fact
only becomes clear by translation into the formal mode of speech,
or, in other words, into syntactical sentences about language and
linguistic expressions.^
Further, the use of the material mode of speech gives rise to
obscurity by employing absolute concepts in place of the syn-
tactical concepts which are relative to language. With regard to
every sentence of syntax, and consequently every philosophical
sentence that it is desired to interpret as syntactical, the lan-
guage or kind of language to which it is to be referred must be
stated. If the language of reference is not given, the sentence is
incomplete and ambiguous. Usually a syntactical sentence is in-
tended to hold in one of the following ways:
1. for all languages ;
2. for all languages of a certain kind ;
3. for the current language of science (or of a sub-domain of
science, such as physics, biology, etc.);
4. for a particular language whose syntactical rules have been
stated beforehand ;
5. for at least one language of a certain kind;
6. for at least one language in general ;
7. for a language (not previously stated) which is proposed as a
language of science (or of a sub-domain of science) ;
8. for a language (not previously stated) whose formulation and
investigation is proposed (apart from the question whether it is to
serve as a language of science or not).
If the formal syntactical mode of speech is used, then linguistic
expressions are being discussed. This makes it quite clear that the
language intended must be stated. In the majority of cases, how-
ever, even if the language is not expressly named, it will be under-
stood from the context which interpretation (say, of those just
given) is intended. The use of the material mode of speech leads, on
the other hand, to a disregard of the relativity to language of philo-
sophical sentences \ it is responsible for an erroneoiis conception of
philosophical sentences as absolute. It is especially to be noted that
the statement of a philosophical thesis sometimes (as in interpreta-
tion 7 or 8) represents not an assertion but a suggestion. Any dis-
pute about the truth or falsehood of such a thesis is quite mistaken,
300
PART V. PHILOSOPHY AND SYNTAX
a mere empty battle of words ; we can at most discuss the utility of
the proposal, or investigate its consequences. But even in cases
where a philosophical thesis presents an assertion, obscurity and
useless controversy are liable to arise through the possibility of
several interpretations (for instance, i to 6). A few examples may
serve to make this clear. (For the sake of brevity, we shall formu-
late these sample theses in a more elementary manner than would
be done in an actual discussion.)
Philosophical sentences Syntactical sentences
(Material mode of speech) (Formal mode of speech)
21 <2. Numbers are classes of 216. Numerical expressions
classes of things. are class-expressions of the
second level.
• 22 a. Numbers belong to a 226. Numerical expressions
special primitive kind of objects, are expressions of the zero-level.
Let us assume that a logicist holds thesis 21a, and a formalist
thesis 22 a. Then between these two there can be endless fruitless
discussion as to which of them is right and what numbers actually
are. The uncertainty disappears as soon as the formal mode of
speech is applied. First of all, theses 21 a and 22 a should be trans-
lated into 21 b and 22 b, But these sentences are not yet complete,
because the statement of the language intended is lacking. Various
interpretations — such, for instance, as those mentioned previously —
are still possible. Interpretation 3 is obviously not intended. Under
interpretation i both parties would be wrong. Under the minimum
interpretation, 6, both would be right, and the controversy would be
at an end; for it is possible to construct a language of arithmetic
either in such a way that 2 1 6 is true or in such a way that 22 b is true.
Perhaps, however, the two disputants agree that they intend their
theses as proposals in the sense of 7, for instance. In that case, the
question of truth or falsehood cannot be discussed, but only the
question whether this or that form of language is the more ap-
propriate for certain purposes.
23 a. Some relations belong to 23 b. Some two- (or more-)
the primitive data. termed predicates belong to the
undefined descriptive primitive
symbols.
24 a. Relations are never primi- 24 b. All two- and more-termed
tive data, they depend upon the predicates are defined on the
properties of their members. basis of the one-termed predi-
cates.
In the case of theses 23 a and 24 a, discussion is again fruitless and
deluded until the disputants pass over to the formal mode of
§78. CONFUSION CAUSED BY THE MATERIAL MODE 3OI
speech and agree as to
for sentences 23 h and 24 6.
25 a. A thing is a complex of
sense-data.
26 a. A thing is a complex of
atoms.
I to 8 is intended
25 b. Every sentence in which
a thing-designation occurs is
equipollent to a class of sen-
tences in which no thing-desig-
nations but sense-data designa-
tions occur.
26 h. Every sentence in which
a thing-designation occurs is
equipollent to a sentence in
whxh space-time co-ordinates
and certain descriptive functors
(of physics) occur.
which of the interpretations
Suppose that a positivist maintains thesis 25 a, and a realist thesis
26 a. Then an endless dispute will arise over the pseudo-question of
what a thing actually is. If we transfer to the formal mode of speech,
it is in this case possible to reconcile the two theses, even if they are
interpreted in the sense of 3, that is, as assertions about the whole
language of science. For the various possibilities of translating a
thing-sentence into an equipollent sentence are obviously not in-
compatible with one another. The controversy between positivism and
realism is an idle dispute about pseudo-theses which owes its origin en-
tirely to the use of the material mode of speech.
Here again we want to emphasize the fact that it does not follow
from the given examples that all sentences of the material mode of
speech are necessarily incorrect. But they are usually incomplete.
Even this does not prevent their correct use ; for in every domain
incomplete, abbreviated modes of speech may frequently be em-
ployed with profit. But the examples show how important it is in
using the material mode of speech, especially in philosophical dis-
cussions, to be fully aware of its character, so as to be able to avoid
the dangers inherent in it. As soon as, in a discussion, obscurities
and doubts of the kind here described arise, it is advisable to
translate at least the principal thesis involved in the controversy
into the formal mode of speech, and to render it more precise by
stating whether it is meant as an assertion or as a suggestion,
and to which language it refers. If the exponent of a thesis
refuses to make these statements concerning it, the thesis is in-
complete and therefore ineligible for discussion.
302
PART V. PHILOSOPHY AND SYNTAX
§ 79. Philosophical Sentences in the Material
AND IN THE FORMAL MODE OF SPEECH
We will now give a series of further examples of sentences in the
material mode of speech, together with their translations into the
formal mode. These are sentences such as commonly occur in
philosophical discussions, sometimes in those of the traditional
sort, sometimes in investigations which are already expressly
oriented in accordance with the logic of science. [For the sake of
brevity, the sentences are, to a certain extent, formulated in a
simplified way.] These illustrative sentences (as also those of § 78)
have not, for the most part, the simple form of those for which we
formulated the criterion of the material mode of speech in an earlier
section. But they have the general feature which is characteristic
of the material mode of speech ; they speak about objects of some
kind, but in such a way that it is possible to construct correlated
sentences of the formal mode of speech which make corresponding
assertions about the designations of these objects. Since the
original sentence, in most cases, cannot be understood univocally, a
particular translation into the formal mode of speech cannot uni-
vocally be given ; it cannot even be stated with certainty that the
sentence in question is a pseudo-object-sentence and, hence, a
sentence of the material mode of speech. The translation given
here is accordingly no more than a suggestion and is in no way
binding. It is the task of anyone who wishes to maintain the
philosophical thesis in question to interpret it by translating it
into an exact sentence. This latter may sometimes be a genuine
object-rsentence (that is to say, not a quasi-syntactical sentence);
and, in that case, no material mode o^ speech occurs. Otherwise it
must be possible to give the interpretation by means of translation
into a syntactical sentence. The syntactical sentences of the fol-
lowing examples — like those of the preceding ones — must further
be completed by stating the language which is referred to; from
this statement it can then be seen whether the sentence is an
assertion or a proposal, e.g. a new rule. We have omitted these
statements in the examples which follow, because as a rule it is
impossible to obtain them univocally from the philosophical sen-
tences of the material mode of speech. [Here, as in the earlier
§ 79* philosophical sentences in the two modes 303
examples, it obviously makes no difference to our investigations
whether the illustrative sentences are true or not.]
Philosophical sentences | Syntactical sentences
(Material mode of speech) | (Formal mode of speech)
A. Generalities (about things, properties, facts, and so on). Here
belong also Examples 7, 9, 17-20.
27 ii. A property of a thing- : 276. A ®pr is not a *pr.
property is not itself a thing- |
property. !
28 a. A property cannot pos- | 286. There is no pr of a level
sess another property. (As op- I higher than the first. (As op-
posed to 27 a.) posed to 276.)
29 a. The world is the totality 29 b. Science is a system of
of facts, not of things. sentences, not of names. ^ •
30 a. A fact is a combination 306. A sentence is a series of
of objects (entities, things). symbols.
31a. If I know an object, 316. If the genus of a symbol
then I also know all the possi- is given, then all the possibilities
bilities of its occurrence in facts, of its occurrence in sentences are
also given.
32 a. Identity is not a relation 326. The symbol of identity
between objects. is not a descriptive symbol.
Sentences 29a to 32 a come from Wittgenstein. Similarly many
other sentences of his which at first appear obscure become clear
when translated into the formal mode of speech.
33a. This circumstance (or: 336. This sentence is ana-
fact, process, condition) is logi- lytic; ...contradictory; ...not
cally necessary; ...logically im- contradictory,
possible (or: inconceivable); ... |
logically possible (or: conceiv- |
able). i
34a. This circumstance (or: 34ft. This sentence is valid;
fact, process, condition) is really . . .contravalid ; ...not contra-
(or: physically, in accordance valid,
with natural laws) necessary ; . , .
really impossible; ...really pos-
sible.
The circumstance (or 356. Si is an L>-consequence
fact, process, condition) Cj is a (or a P-consequence, respec-
logically (or really) necessary tively) of Sg.
condition for the circumstance
C,.
33 a to 35 a are sentences of modality ; see § 69.
304
PART V. PHILOSOPHY AND SYNTAX
36 a. A property of an object
c is called an essential (or: in-
ternal) property of c, if it is in-
conceivable that c should not
possess it (or: if c necessarily
possesses it); otherwise it is an
inessential (or : external) property.
(Correspondingly for a relation.)
366. pti is called an analytic
(or, if desired: an essential or
internal) predicate in relation
to an object-designation if
P^i(^i) is analytic. (Correspond-
ingly for a two- or more-tenned
predicate.'!
The uncertainty of the formulation 36 a is shown by the fact that
it leads to obscurities and contradictions. Let us take as the object
c, for example, the father of Charles. According to definition 36^2,
being related to Charles is an essential property of c, since it is in-
conceivable that the father of Charles should not be ^related to
Charles. But being a landowner is not an essential property of the
father of Charles. For, even if he is a landowner, it is conceivable
that he might not be one. On the other hand, being a landowner is
an essential property of the owner of this piece of land. For it is
inconceivable that the owner of this piece of land should not be a
landowner. Now, however, it happens to be the father of Charles
who is the owner of this piece of land. On the basis of definition
36 a, it has just been proved that it is both an essential and not an
essential property of this man to be a landowner. Thus 36 a leads
to a contradiction; but 366 does not, because ‘landowner* is an
analytic predicate in relation to the object-designation ‘ the owner of
this piece of land ’, but it is not an analytic predicate in relation to
the object-designation ‘ the father of Charles *. Hence the fault of
definition 36 a lies in the fact that it is referred to the one object in-
stead of to the object-designations^ which may be different even when
the object is the same.
This example shows (as will easily be confirmed by a closer in-
vestigation) that the numerous discussions and controversies about
external and internal properties and relations are idle, if, as is usual,
they are based on a.definition of either the form indicated or one re- ’
sembling it, or, at any rate, on one which is formulated in the material
mode of speech. [Such investigations are esp^-, 'tally to be found in
the work of Anglo-Saxon philosophers, and it was through them
that Wittgenstein, although it is to him that we owe the detection of
many other pseudo-questions, was himself misled into enquiries of
this nature.] If instead of the usual sort of definition, a definition in
the formal mode is given, then the situation in these commonlv dis-
puted cases becomes unambiguous, and moreover so simple that no
one can any longer be tempted to raise philosophical problems about it.
B. The so-called philosophy of number; logical analysis of arithmetic.
Here belong also Examples 10, 17, zi, and 22.
37 a. God jcreated the natural 37 b. The natural-number
numbers (integers); fractions symbols are primitive symbols;
§ 79 * philosophical SENTENCES IN THE TWO MODES 305
and real numbers, on the other
hand, are the work of man.
(Kronecker.)
38 a. The natural numbers are
not given ; only an initial term of
the process of counting and the
operation of progression from
one term to the next are given;
the other terms are created pro-
gressively by means of this
operation.
39 a. The mathematical con-
tinuum is a series of a certain
structure ; the terms of the series
are the real numbers.
40 a. The mathematical con-
tinuum is not composed of
atomic elements, but is a whole
which is analysable into ever
further analysable sub-intervals.
A real number is a series of in-
tervals contained one inside the
other.
the fractional expressions and
the real-number expressions are
introduced by definition.
38 b. The natural number ex-
pressions are not primitive sym-
bols (as opposed to 376); only
‘ 0 ’ and ‘ • ’ are primitive sym-
bols ; an St has the form nu or
St*. (Languages I and II.)
29 b. A to which certain
structural properties (density,
continuity, etc.) are attributed in
the axioms, is a primitive sym-
bol. The arguments which are
suitable to ptj — they are expres-
sions of the zero-level— are
called real-number expressions.
406. A p i \ , to which certain
structural properties (namely,
those of a part- whole relation of a
certain kind) are attributed in the
axioms, is a primitive symbol. An
gu^ whose arguments are natural-
number expressions and whose
value-expressions are suitable as
arguments to pti is called a real-
number expression. [A so-called
creative sequence of selections is
then represented by an glib ; see
p. 148.]
39 a and 40 « present (in a simplified formulation) the antithesis
between the usual mathematical conception of the continuum of real
numbers, based on the theory of aggregates, and the intuitionist con-
ception of the continuum represented by Brouwer and Weyl, which
rejects the former as atomistic. 39 b and 40 b may be interpreted as
suggestions for the construction of two different calculi.
C. Problems of the so-called given or primitive data {epistemology,
phenomenology); logical analysis of the protocol sentences.
Here belong also Examples 23 and 24.
41a. The only primitive data 416. Only two- or more-
are relations between experi- termed predicates whose argu-
ences. ments belong to the genus of the
experience-expressions occur as
descriptive primitive symbols.
PART V. PHILOSOPHY AND SYNTAX
306
42 a. A temporal series of
visual fields is given as primitive
data ; every visual field is a two-
dimensional system of positions
which are occupied by colours.
(As opposed to 41 a.)
43 a. The sense-qualities, such
as colours, smells, etc., belong to
the primitive data.
44 a. The fact that the system
of colours arranged according to
similarity (the so-called colour-
pyramid) is three-dimensional, is
known a priori (or : is to be ap-
prehended by intuition of es-
sence ; or : is an internal property
of that arrangement).
45 a. The colours are not
originally given as members of
an order, but as individuals; an
empirical relation of similarity
exists between them, however,
on the basis of which the colours
can be arranged empirically in a
three-dimensional order.
426. A descriptive atomic
sentence consists of a time co-
ordinate, two space co-ordinates
and a colour expression.
43 h. Symbols of sense-quali-
ties, such as colour-symbols,
smell-symbols, etc., belong to
the descriptive primitive sym-
bols.
44 6. A colour-expression
consists of three co-ordinates ; the
values of each co-ordinate form a
serial order according to syn-
tactical rules; on the basis of
these syntactical rules, therefore,
the colour-expressions consti-
tute a three-dimensional order.
45 A. The colour expressions
are not compound ; they are
primitive symbols ; further, a
symmetrical, reflexive, but not
transitive, prj to which the
colour-expressions are suitable
as arguments, occurs as a primi-
tive symbol ; the theorem of the
three-dimensionality of the order
determined by this pr is P-
valid.
The much-disputed philosophical question as to whether the
knowledge of the three'-dimensionality of the colour-pyramid is a priori
or empirical is thus, by reason of the use of the material mode of
speech, incomplete. The answer is dependent upon the form of the
language.
46 u. Every colour possesses
three components: colour-tone,
saturation, and intensity (or:
colour-tone, white-content, and
black-content).
47 a. Every colour is at a place.
48 a. Every tone has a certain
pitch.
46 b. Every colour-expression
consists of three partial expres-
sions (or ; is synonymous with an
expression composed in this
way): one colour-tone expres-
sion, one saturation-expression,
and one intensity-expression.
476. A colour-expression is
always accompanied in a sen-
tende by a place-designation.
48 6. Every tone-expression
contains an expression of pitch.
§ 79 - philosophical SENTENCES IN THE TWO MODES 307
D. The so-caUed natural phildsophy; logical analysis of the natural
sciences.
Here belong also Examples ii, 25, and 26.
49 a. TVme is continuous. | 496. The real-number ex-
pressions are used as time-
co-ordinates.
See Wittgenstein on this point {[TractatiU] p. 172): “All proposi-
tions such as the law of causation, the law of continuity in nature, . . .
are a priori intuitions of the possible forms of the propositions of
science. ** (Instead of “ a priori intuitions of *’ we would prefer to say :
“ conventions concerning **.)
50^2. is one-dimensional; $ob. A time-designation con-
space is three-dimensional. sists of one co-ordinate ; a space-
designation consists of three co-
ordinates.
51a. Time is infinite in both 51 ft. Every positive or nega-
directions, forwards and back- tive real-number expression
wards. can be used as a time-co-
ordinate.
The opposition between the determinism of classical physics and
the probability determination of quantum physics concerns a syn-
tactical difference in the system of natural laws, that is, of the P-rules
of the physical language (already formulated or still to seek) ; this is
shown by the two following examples.
52 a. Every process is uni- 52 ft, For every particular
vocally determined by its causes, physical sentence ©j there is, for
any time-co-ordinate which
has a smaller value than the
time-co-ordinate which occurs in
©1, a class Ri of particular sen-
tences with as time-co-
ordinate, such that ©1 is a P-
consequence of
53 a. The position and velo- 53ft. If ©1 is a particular sen-
city of a particle is not univocally tence concerning particles and
but only probably determined by '2li a time-co-ordinate of smaller
a previous constellation of par- value than that which occurs in
tides. ©1, then ©j is not a P-conse-
quence of a class of such sen-
tences with 'III as time-co-
ordinate, however comprehen-
sive, but only a probability-
consequence of such a class with
a coefficient of probability smal-
ler than I.
20-2
3o8
PART V. PHILOSOPHY AND SYNTAX
§ 8o. The Dangers of the Material Mode
OF Speech
If we wish to characterize the material mode of speech by one
general term, we may say, for instance, that it is a special kind of
transposed mode of speech. By a transposed mode of speech we mean
one in which, in order to assert something about an object a, some-
thing corresponding is asserted about an object h which stands in a
certain relation to the object a (this does not pretend to be an exact
definition). For example, every metaphor is a transposed mode of
speech ; but other kinds also occur frequently in ordinary language
— far more frequently than one may at first believe. The use of a
transposed mode of speech can easily lead to obscurities ; but when
systematically carried into effect, it is non-contradictory.
Examples of different kinds of transposed mode of speech,
I. An artificial example. The tenn ‘marge* (as a term parallel to
‘ large ’) is introduced by means of the following rule : if a place has
more than 10,000 inhabitants, then we shall say that the place
whose name precedes that of a in the alphabetical list of places, is
marge. A rule of this kind can be carried into effect without any
contradiction ; for instance, according to it, the place Berlichingen is
marge, since, in the alphabetical list of places, its name is followed
by ‘Berlin*. The definition seems absurd, since it makes no dif-
ference to the properties (in the ordinary sense) of a place whether
it is marge or not. But the same thing holds for the ordinary
material mocfe of speech also (see below. Example 5), even (as one
finds on examination, in opposition, of course, to the view commonly
held) for Examples 2, 3, and 4. 2. According to the ordinary use
of language, a man is called famous if other people make asser-
tions of a certain kind about him. 3. According to the ordinary use
of language, an action a of a certain person is called legal crime if the
penal law of the country in which that person lives places the de-
scription of a kind of action to which a belongs in the list of crimes.
4. According to the ordinary use of language, an action a of a certain
person is called a moral crime if, in the minds of the majority of other
persons, the thought of someone (but not themselves) committing
an action of this kind calls forth the feeling of moral indignation.
5. According to the ordinary use of language, it is said of a city (for
instance, of Babylon ; see the example in § 74) that it has been
treated of in a certain lecture (material mode of speech) if a designa-
tion of the city has occurred in this lecture. For the qualities (in the
ordinary sense) of the city in question, it is not of the least importance
whether it has* the property of having been treated of in yesterday*s
lecture or not. This property is therefore a transposed property.
§ 8o. DANGERS OF THE MATERIAL MODE 309
The material mode of speech is a transposed mode of speech.
In using it, in order to say something about a word (or a sentence)
we say instead something parallel about the object designated by
the word (or the fact described by the sentence, respectively). The
origin of a transposed^mode of speech can sometimes be explained
psychologically by the fact that the conception of the substituted
object h is for some reason more vivid and striking, stronger in
feeling-tone, than the conception of the original object a. This is
the case with the material mode of speech. The image of a word
(for instance, of the word ‘ house *) is often much less vivid and
lively than that of the object which the word designates (in the
example, that of the house). Further, the fact, which is perhaps a
consequence of the psychological fact just mentioned, that the
approach and method of syntax have hitherto not been sufficiently
known, and that, in consequence, the majority of the necessary
syntactical terms have not been a part of ordinary language, may
have contributed to the origin of the material mode of speech. For
this reason, instead of saying: “The sentence "a has three books,
h has two books, and a and b together have seven books ’ is contra-
dictory ”, we say: “ It is impossible (or inconceivable) for a to have
three books, b two books, and a and b together seven books”; or
(which has an even stronger resemblance to an object-sentence):
“If a has three books, and b two, then a and b together cannot
possibly have seven books.” People are not accustomed to direct
their attention to the sentence instead of the fact; and it is ap-
parently much more difficult to do so. In addition, there is the
circumstance that, in ordinary language, we have no syntactical
expression which is equivalent in meaning to ‘contradictory*,
while the quasi-syntactical expression ‘impossible* is ready to
hand.
How difficult it is even for scientists to adopt the syntactical point
of view, that is to say, to pay attention to the sentences instead of to
the facts, is shown especially clearly in the typical misunderstand-
ings which one encounters again and again in discussing logical
questions even with scientists, and still more with philosophers.
For instance, when we of the Vienna Circle criticize, in accordance
with our anti-metaphysical view, certain sentences of metaphysics
(such as: “There is a God”) or of metaphysical epistemology (such
as : “ The external world is real ”) we are interpreted by the majority
of our opponents as denying those object-sentences and conse-
quently affirming others (such as: “There is no God” or: “The ex-
PART V. PHILOSOPHY AND SYNTAX
310
temal world is not real etc.). These misunderstandings are always
occurring in spite of the fact that we have already explained them
many times (sec, for instance, Carnap [Scheinprohleme\, Schlick
[Positivismus\y Carnap [Metaphyiik\)y and are constantly pointing
out that we are not talking about the (supposititious) facts, but about
the (supposititious) sentences; in the mode of expression of this
book: the thesis maintained by us is not an object-sentence but a
syntactical sentence.
The suggestions we have given are intended only to throw light
upon, and not by any means to answer, the question of the psycho-
logical explanation of transposed modes of speech in general, and
of the material mode in particular. To investigate it more closely
would be well worth while; but we must leave that task to the
psychologists. What we must here take into account is the fact that
th«5 material mode of speech is a part of ordinary linguistic usage,
and that it will continue to be frequently employed, even by our-
selves. Therefore it behoves us to pay special attention to the
dangers connected with its use.
Most of the ordinary formulations in the material mode of
speech depend upon the use of universal words. Universal words
very easily lead to pseudo-problems \ they appear to designate kinds
of objects, and thus make it natural to ask questions concerning the
nature of objects of these kinds. For instance, philosophers from
antiquity to the present day have associated with the universal
word * number' certain pseudo-problems which have led to the
most abstruse inquiries and controversies. It has been asked, for
example, whether numbers are real or ideal objects, whether they
are extra-mental qr only exist in the mind, whether they are the
creation of thought or independent of it, whether they are potential
or actual, whether real or fictitious. The question of the origin of
numbers has been raised, and has been found to be due to a division
of the self, to an original primitive intuition of duality in unity, and
so forth. Similarly, innumerable questions have been put con-
cerning the nature of space and timey not only by speculative meta-
physicians (up to recent times), but also by many philosophers
whose epistemological theses are ostensibly (as with Kant)
oriented in accordance with empirical science. opposed to all
this, an inquiry \vhich is free from metaphysics and concerned with
the logic of science can only have as its object the syntax of the
spatio-temporal expressions of the language of science, in the
§8o. DANGERS OF THE MATERIAL MODE 3II
form, say, of an axiomatics of the space-time system of physics (as,
for instance, the researches of Reichenbach [Axiomatik]). Further,
mention should be made of the many pseudo-problems concerning
the nature of the physical and the psychical. Again, the pseudo-
questions concerning properties and relations and with them the
whole controversy about universals rests on the misleading use of
universal words. All pseudo -questions of this kind disappear if the
formal instead of the material mode of speech is used, that is, if in
the formulation of questions, instead of universal words (such as
‘number*, ‘space*, ‘universal*), we employ the corresponding
syntactical words (‘numerical expression*, ‘space-co-ordinate*,
‘predicate’, etc.).
We have already met with a number of examples in which the
use of the material mode of speech leads to contradictions. The
danger of the occurrence of such contradictions is especially ^reat
in the case of languages which are mutually translatable, or, from
the standpoint of one language of science, of two sub-languages
between the sentences of which certain relations of equipollence
(not necessarily of L-equipollence) hold. This applies, for in-
stance, to the language of psychology and the language of physics.
If the material mode of speech is employed in relation to the psy-
chological language (by the use, for instance, of universal words
like ‘the psychical*, ‘psyche’, ‘psychical process*, ‘mental pro-
cess*, ‘act*, ‘experience*, ‘content of experience*, ‘intentional
object *, and so on), and if, in the same investigation, it is also used
in relation to the physical language (either the everyday language
or the scientific language), hopeless confusion frequently ensues.
The danger here indicated has been described by us in detail on
other occasions {\Phys. Sprache] pp. 453 ff., [Unity]). Compare also
[Psychol.] p. 186, where attention is drawn to the obscurities which
arise from the use of the material mode of speech in the sentences
of a psychologist; further, see [Psychol.] p. 181 for the origin of a
pseudo-problem due to the material mode of speech. The examples
on p. 314 under I also belong in part here. On the psycho-physical
problem, see p. 324.
From the earlier examples, which could easily be multiplied, it
is clear that the use of the material mode of speech often gives rise
to an obscurity, an ambiguity, which is manifested, for instance,
in the fact that essentially different translations into the formal
mode of speech are possible. In more extreme cases, contradic-
312
PART V. PHILOSOPHY AND SYNTAX
tions also appear. These contradictions are, however, frequently
not at all obvious, for the reason that the consequences are not de-
rived by means of formal rules, but by means of material con-
siderations, in which it is often possible to avoid the traps that
one has set oneself by this dubious formulation. Even where no
contradictions or ambiguities occur, the use of the material mode
of speech has the disadvantage of leading easily to self-deception as
regards the object under discussion: one believes that one is in-
vestigating certain objects and facts, whereas one is, in reality,
investigating their designations, i.e. words and sentences.
§ 8i. The Admissibility of the Material
Mode of Speech
We have spoken of dangers and not of errors of the material
mode of speech. The material mode of speech is not in itself erroneous ;
it only readily lends itself to wrong use. But if suitable definitions
and rules for the material mode of speech are laid down and
systematically applied, no obscurities or contradictions arise.
Since, however, the word-language is too irregular and too com-
plicated to be actually comprehended in a system of rules, one
must guard against the dangers of the material mode of speech as
it is ordinarily used in the word-language by keeping in mind the
peculiar character of its sentences. Especially when important
conclusions or philosophical problems are to be based on sentences
of the material mode of speech, it is wise to make sure of their
freedom from ambiguity by translating them into the formal mode.
It is not hy any means suggested that the material mode of speech
should he entirely eliminated. For since it is established in general
use, and is thus more readily understood, and is, moreover, often
shorter and more obvious than the formal mode, its use is fre-
quently expedient. Even in this book, and especially in this Part,
the material mode of speech has often been employed; here are
some examples :
Material mode of speech Formal mode of speech
54 a. Philosophical questions 546. In philosophical ques-
are sometimes concerned with tions expressions sometimes oc-
objectswhichrdo not occur in the cur which do not occur in the
object-domain of the empirical languages of the sciences; for
§8l. ADMISSIBILITY OF THE MATERIAL MODE 313
sciences. For example: the example, the expressions : ‘ thing-
thing-in-itself, the transcen- in-itself\ ‘the transcendental*,
dental, and the like (p. 278). etc.
55 a. An object-question is 55 b. In an object-question,
concerned, for instance, with the predicates of the language of
properties of animals ; on the zoology (designations of kinds of
other hand, a logical cfuestion is animals) occur ; on the other
concerned with the sentences of hand, in a logical question, de-
zoology (p. 278). signations of sentences of the
zoological language occur.
56 a. It is just as easy to con- 566. It is just as easy to con-
struct sentences about the forms struct sentences in which, as
of linguistic expressions as it is to predicates, syntactical predicates
construct sentences about the occur, and, as arguments, syn-
geometrical forms of geometrical tactical designations of expres-
structures (pp. 282 f.). sions, as it is to construct sen-
tences in which, as predicates,
predicates of the language of
(pure) geometry occur, and, as
arguments, object-4esignations
of the language of geometry.
If a sentence of the material mode of speech is given, or, more
generally, a sentence which is not a genuine object-sentence, then
the translation into the formal mode of speech need not always be
undertaken, but it must always be possible. Translatability into the
formal mode of speech constitutes the touchstone for all philosophical
sentences, or, more generally, for all sentences which do not belong
to the language of any one of the empirical sciences. In in-
vestigating translatability, the ordinary use of language and the
definitions which may have been given by the author must be taken
into consideration. In order to find a translation, we attempt to
use, wherever a universal word occurs (such as ‘ number * or ‘ pro-
perty ’) the corresponding syntactical expression (such as ‘ numeri-
cal expression ’ or ‘ property-word *, respectively). Sentences
which do not, at least to a certain extent, univocally determine their
translation are thereby shown to be ambiguous and obscure.
Sentences which do not give even a slight indication to determine
their translation are outside the realm of the language of science
and therefore incapable of discussion, no matter what depths or
heights of feeling they may stir. Let us give a few warning ex-
amples of such sentences as they occur in the writings of our own
circle or in those of closely allied authors. The majority of readers
314
PART V. PHILOSOPHY AND SYNTAX
will scarcely, I think, succeed in finding a translation of these into
the formal mode of speech that would satisfactorily represent the
author’s meaning. Even if the author himself is perhaps able to
give such a translation — and in some cases even this seems doubtful
— his readers will certainly fall into confusion and uncertainty.
We shall see that the sentences in which the word ‘ inexpressible ’
or something similar occurs are especially dangerous. In the
examples under heading I we find a mythology of the inexpressible ^
in the examples under II a mythology of higher things^ and in
Sentence 13 both of these.
I. I. There is indeed the inexpressible. 2. The qualities which
appear as content of the stream of consciousness can neither be as-
serted, described, expressed, nor communicated, but can only be
manifested in experience. 3. What can be shown cannot be said.
4. The given experience possesses an utterable structure, but at the
same time it possesses an unutterable content which is nevertheless
very well known to us. 5. Human beings must verify psychological
sentences by their own unutterable experience, which is nevertheless
very well known to them ; they must examine whether the sentence
in question, the combination of symbols, is isomorphous (like in
structure) with their unutterable experience. 6. The unutterable
experience blue or bitter,... 7, The essence of individuality cannot
be represented in words, and is indescribable, and therefore meaning-
less for science. 8. Philosophy will mean the unspeakable by clearly
displaying the speakable. 9. The holding [subsistence] of [formal or]
internal properties and relations cannot be asserted by propositions
[sentences]. '
II . 10. The sense of the world must lie outside the world.
I T . How the world is, is completely indifferent to what is higher.
12. If good or bad willing changes the world it can only change the
limits of the world, not the facts. 13. Propositions [sentences] can-
not express anything higher.
Let us suggest a few possibilities of translation which, however,
probably do not correspond to the intentions of the authors. In the
case of Sentence i it would be necessary to distinguish between two
interpretations: i A. “There are unutterable objects”, that is to say,
“There are objects for which no object-designations exist”; trans-
latio 7 i: “There are object-designations which are not object-desig-
nations.” I B. “ There are unutterable facts ”, that is to say, “ There
are facts which are not described by any sentence”; translation:
“There are sentences which are not sentences.” Concerning 6:
in other words, “The experience designated by the word ‘blue’
cannot be designated by any word”; translation: “The experience-
designation -blue’ is not an experience-designation.” Sentence 9
means : “The fact that a property of a certain kind appertains to an
§8l. ADMISSIBILITY OF THE MATERIAL MODE 315
object cannot be asserted by means of a sentence’*; translation:
“A sentence in which a property- word of a certain kind occurs is
not a sentence.” Sentence 13 means: “The higher facts cannot be
expressed by means of sentences”; translation: higher sen-
tences are not sentences.”
Let it be once more 'called to mind that the distinction between
the formal and the material modes of speech does not refer to
genuine object-sentences and therefore not to the sentences of the
empirical sciences, or to sentences of this kind which occur in the
discussions of the logic of science (or of philosophy). (See the
three columns, on p. 286.) It is here a question of the sentences of
the proper logic of science. According to the ordinary use of
language it is customary to formulate these partly in the fonn of
logical sentences and partly in the form of object-sentences. Our
investigations have shown that the supposititious object-senteAces
of the logic of science are pseudo-object-sentences, or sentences
which apparently speak about objects, like the real dbject-sen-
tences, but which in reality are speaking about the designations of
these objects. This implies that all the sentences of the logic of
science are logical sentences; that is to say, sentences about lan-
guage and linguistic expressions. And our investigations have
further shown that all these sentences can be fomiulated in such a
way as to refer not to sense and meaning but to the syntactical form
of the sentences and other expressions — they can all be translated
into the formal mode of speech, or, in other words, into syntactical
sentences. The logic of science is the syntax of the language of science.
B. THE LOGIC OF SCIENCE AS SYNTAX
§ 82. The Physical Language
The logical analysis of physics — as a part of the logic of science
— is the syntax of the physical language. All the so-called epi-
stemological problems concerning physics (in so far as it is not a
question of metaphysical pseudo-problems) are in part empirical
questions, the majority of which belong to psychology, and in part
logical questions which belong to syntax. A more exact exposition
of the logical analysis of physics as the syntax of the physical Ian-
3i6 party, philosophy and syntax
guage must be left for a special investigation. Here we shall only
offer a few suggestions towards it.
The logical analysis of physics will have, in the first place, to
formulate rules of formation for sentences and other kinds of ex-
pressions of the physical language (see § 40). The most important
expressions which occur as arguments are the point-expressions
(designations of a spatio-temporal point, consisting of four real-
number expressions, namely, three space-co-ordinates and one
time-co-ordinate) and the domain-expressions (designations of a
limited space-time domain). The physical coefficients of states are
represented by descriptive functors. The descriptive functors and
predicates can be divided into those having point-expressions and
those having domain -expressions as arguments.
The sentences can be classified according to their degree of
generality. We will here only discuss the two extreme kinds of sen-
tences and, for the sake of simplicity, only those in which all the
interior arguments are point- or domain-expressions : the concrete
sentences contain no unrestricted variables; the laws contain no
constants as interior arguments.
Either L-rules alone, or L-rules and P-rules, can be laid down as
transformation rules of the physical language. If P-rules are desired,
they will generally be stated in the form of P-primitive sentences.
In the first place, certain most general laws will be formulated as
P-primitive sentences ; we will call these primitive laws. In addi-
tion, descriptive synthetic sentences of another form — even con-
crete ones — may be stated as P-primitive sentences. In the ma-
jority of cases, the primitive laws will have the form of a universal
sentence of implication or of equivalence. The primitive laws and
the other valid laws can be either deterministic or laivs of proh^
ability ; the latter can be formulated, for instance, with the help of
a probability implication. Since the concept of probability is a very
significant one for physics, particularly in view of the latest de-
velopments, the logical analysis of physics will have thoroughly to
investigate the syntax of the sentences of probability ; and it may
be found possible to establish a connection with the concept of
range in the general syntax.
We cannot go more fully into the concept of probability here. See
the lectures and discussions of the Prague Congress {Erkenntnis
I, 1930); further bibliographical references are given in Erkenntnis
§82. THE PHYSICAL LANGUAGE 317
II, 189 f., 1931 ; there are also investigations, as yet unpublished, by
Reichenbach, Hempel, and Popper.* On the probability implica-
tion, see Reichenbach [Wahrscheinlichkeitslogik].
Syntactical rules will have to be stated concerning the forms
which the protocol-sentencesy by means of which the results of ob-
servation are expressed, may take. [On the other hand, it is not the
task of syntax to determine which sentences of the established
protocol fbrm are to be actually laid down as protocol-sentences,
for ‘ true * and ‘ false * are not syntactical terms ; the statement of the
protocol-sentences is the affair of the physicist who is observing
and making protocols. ]
A sentence of physics, whether it is a P-primitive sentence, some
other valid sentence, or an indeterminate assumption (that is, a
premiss whose consequences are in course of investigation), will l^e
tested by deducing consequences on the basis of the transformation
rules of the language, until finally sentences of the form of protocol-
sentences are reached. These will then be compared 'with the
protocol-sentences which have actually been stated and either con-
firmed or refuted by them. If a sentence which is an L-conse-
quence of certain P-primitive sentences contradicts a sentence
which has been stated as a protocol-sentence, then some change
must be made in the system. For instance, the P-rules can be
altered in such a way that those particular primitive sentences are
no longer valid; or the protocol-sentence can be taken as being
non-valid ; or again the L-niles which have been used in the de-
duction can also be changed. There are no established rules for the
kind of change which must be made.
Further, it is not possible to lay down any set rules as to how
new primitive laws are to be established on the basis of actually
stated protocol-sentences. One sometimes speaks in this connection
of the method of so-called induction. Now this designation may be
retained so long as it is clearly seen that it is not a matter of a regular
method but only one of a practical procedure which can be
investigated solely in relation to expedience and fruitfulness. That
there can be no rules of induction is shown by the fact that the
L-content of a law, by reason of its unrestricted universality,
always goes beyond the L-content of every finite class of protocol-
• {Note, 1935.) These works have meantime appeared; see
Bibliography.
3i8 party, philosophy and syntax
sentences. On the other hand, exact rules for deduction can be
laid down, namely, the L-rules of the physical language. Thus the
laws have the character of hypotheses in relation to the protocol-
sentences; sentences of the form of protocol-sentences may be
L-consequences of the laws, but a law cannpt be an L-consequence
of any finite synthetic class of protocol-sentences. The laws are not
inferred from protocol-sentences, but are selected and laid down
on the grounds of the existing protocol-sentences, which are always
being re-examined with the help of the ever-emerging new protocol-
sentences. Not only laws, however, but also concrete sentences
are formulated as hypotheses, that is to say, as P-primitive sen-
tences — -such as a sentence about an unobserved process by which
certain observed processes can be explained. There is in the strict
sense no refutation (falsification) of an hypothesis ; for even when
it proves to be L-incompatible with certain protocol-sentences,
there always exists the possibility of maintaining the hypothesis
and renouncing acknowledgment of the protocol-sentences. Still
less is there in the strict sense a complete confirmation (verifica-
tion) of an hypothesis. When an increasing number of L-conse-
quences of the hypothesis agree with the already acknowledged
protocol -sentences, then the hypothesis is increasingly confirmed;
there is accordingly only a gradually increasing, but never a final,
confirmation. Further, it is, in general, impossible to test even a
single hypothetical sentence. In the case of a single sentence of
this kind, there are in general no suitable L-consequences of the
form of protocol-sentences ; hence for the deduction of sentences
having the form of protocol-sentences the remaining hypotheses
must also be used. Thus the test applies^ at bottom y not to a single
hypothesis but to the whole system of physics as a system of hypotheses
(Duhem, Poincar^).
No rule of the physical language is definitive; all rules are laid
down with the reservation that they may be altered as soon as it
seems expedient to do so. This applies not only to the P-rules but
also to the L-rules, including those of mathematics. In this re-
spect, there are only differences in degree ; certain rules are more
difficult to renounce than others. [If, however, we assume that
every new protocol-sentence which appears within a language is
synthetic, there is this difference between an L-valid, and there-
fore analytic, sentence and a P-valid sentence namely, that
§82. THE PHYSICAL LANGUAGE 319
such a new protocol-sentence- — independently of whether it is
acknowledged as valid or not — can be, at most, incompatible with
S2 but never with Si- In spite of this, it may come about that,
under the inducement of new protocol-sentences, we alter the
language to such an extent that is no longer analytic.]
If a new V -primitive sentence Si is stated, but without sufficient
transformation rules by which, from Si in conjunction with the
other P-primitive sentences, sentences of the form of protocol-
sentences could be deduced, then in principle Si cannot be tested,
and is therefore useless from the scientific point of view. If, how-
ever, sentences of the form of protocol-sentences are deducible
from Si in conjunction with the remainder of the P-primitive
sentences, but only such as are deducible from the remaining
P-primitive sentences alone, then Si as a primitive sentence is un-
productive, and scientifically superfluous.
A new descriptive symbol which is to be introduced need not be
reducible by means of a chain of definitions to symbols which
occur in protocol-sentences. A symbol of this kind may also be
introduced as z. primitive symbol by means of new P-primitive sen-
tences. If these primitive sentences are testable, i.e. if sentences
of the form of protocol-sentences are deducible from them, then
thereby the primitive symbols are reduced to symbols of the
protocol-sentences.
Example: Let protocol-sentences be the observation sentences of
the usual form. The electric field vector of classical physics is not
definable by means of the symbols which occur in such protocol-
sentences; it is introduced as a primitive symbol by the Maxwell
equations which are formulated as P-primitive sentences. There is
no sentence equipollent to such an equation, which contains only
symbols of the protocol-sentences, although, of course, sentences of
protocol form can be deduced from the Maxwell equations in con-
junction with the other primitive sentences of classical physics ; in
this way, the Maxwell theory is empirically tested. Counter-example .
The concept of entclechy ”, employed by the neo-vitalists, must be
rejected as a pseudo-concept. It is, however, not a sufficient justifi-
cation for this rejection to point out that no definition of that concept
is given by means of which it could be reduced to the terms of the obser-
vation sentences ; for the same thing is also true of a number of abstract
physical concepts. The decisive point is rather the fact that no laws
which can be empirically tested are laid down for that concept.
The explanation of a single known physical process, the deduc-
tion of an unknown process in the past or in the present, from one
320 PART V. PHILOSOPHY AND SYNTAX
that is known, and the prediction of a future event, are all operations
of the same logical character. In all three cases it is, namely, a
matter of deducing the concrete sentence which describes the
process from valid laws and other concrete sentences. To explain
a law (in the material mode of speech : a universal fact) means to
deduce it from more general laws.
The construction of the physical system is not effected in ac-
cordance with fixed rules ^ but by means of conventions. These cpn-
ventions, namely, the rules of formation, the L-rules, and the
P-rules (hypotheses), are, however, not arbitrary. The choice of
them is influenced, in the first place, by certain practical methodo-
logical considerations (for instance, whether they make for sim-
plicity, expedience, and fruitfulness in certain tasks). This is the
case for all conventions, including, for example, definitions. But
in addition the hypotheses can and must be tested by experience,
that is to say, by the protocol-sentences — both those that are
already stated and the new ones that are constantly being added.
Every hypothesis must be compatible with the total system of
hypotheses to which the already recognized protocol-sentences
also belong. That hypotheses, in spite of their subordination to
empirical control by means of the protocol-sentences, nevertheless
contain a conventional element is due to the fact that the system
of hypotheses is never univocally determined by empirical material,
however rich it may be.
Let us make brief mention of two theses held by us, upon which,
however, the above view regarding the physical language does not
depend. The thesis of physicalism maintains that the physical lan-
guage is a universal language of science — that is to say, that every
language of any sub -domain of science can be equipollently trans-
lated into the physical language. From this it follows that science
is a unitary system within which there are no fundamentally
diverse object-domains, and consequently no gulf, for example,
between natural and psychological sciences. This is the thesis of
the unity of science. We will not examine these theses in greater
detail here. It is easy to see that both are theses of the syntax of
the language of science.
On the view of the physical language here discussed and on the
theses of physicalism and of the unity of science, see Neurath
\Physicalism\f [Physikalismus’l, [Soziol. Phys,']y \Protohollsdtze\y
§82. THE PHYSICAL LANGUAGE 32 I
iPsycholJ\ \ Carnap [Phys. Sprache\y [PsychoL]^ {Protokollsdtze\, In
the discussions of the Vienna Circle, Neurath has been conspicuous
for his early — often initiatory — and especially radical adoption of
new theses. For this reason, although many of his formulations are
not unobjectionable, he has had a very stimulating and fruitful in-
fluence upon its investigations; for instance, in his demand for a
unified language whicA should not only include the domains of
science but also the protocol-sentences and the sentences about
sentences ; in his emphasis on the fact that all rules of the physical
language depend upon conventional decisions, and that none of its
sentences — not even the pro tocc 1-sentences — can ever be definitive;
and, finally, in his rejection of so-called pre-linguistic elucidations
and of the metaphysics of Wittgenstein. It was Neurath who sug-
gested the designations “ Physicalism ** and “Unity of science”. —
One of the most important problems of the logical analysis of physics
is that of the form of the protocol-sentences and of the operation of
testing (problem of verification) ; on this point, see also Popper.
On the view here expounded the domain of the scientific senteAccs
is not so restricted as on the one formerly held by the Vienna Circle.
It was originally maintained that every sentence, in orde^ to be sig-
nificant, must be completely verifiable (Wittgenstein; Waismann
[Wahrscheinlichkeii] p. 229; and Schlick \Kausalitdt\ p. 150); every
sentence therefore must be a molecular sentence formed of concrete
sentences (the so-called elementary sentences) (Wittgenstein \Trac~
tatus] pp. 102, 1 18 ; Carnap [Aufbau^). On this view there was no place
for the laws of nature amongst the sentences of the language. Either
these laws had to be deprived of their unrestricted universality and
be interpreted merely as report-sentences, or they were left their
unrestricted universality, and regarded not as proper sentences of
the object-language, but merely as directions for the construction of
sentences (Ramsey {Foundations] pp. 237 ff. ; Schlick [Kausalitdt]
pp. i5of., with references to Wittgenstein), and hence as a kind of syn-
tactical rules. In accordance with the principle of tolerance, we will
not say that a construction of the physical language corresponding
to this earlier view is inadmissible ; it is equally possible, however, to
construct the language in such a way that the unrestrictedly universal
laws are admitted as proper sentences. The important difference
between laws and concrete sentences is not obliterated in this
second form of language, but remains in force. It is taken into
account in the fact that definitions are framed for both kinds of sen-
tences, and their various syntactical properties are investigated. The
choice between the two forms of language is to be made on the
grounds of expedience. The second form, in which the laws are
treated as equally privileged proper sentences of the object-language,
is, as it appears, much simpler and better adapted to the ordinary use
of language in the actual sciences than the first form. A detailed
criticism of the view according to which laws are not sentences is
given by Popper.
SL
22
322
PART V. PHILOSOPHY AND SYNTAX
The view here presented allows great freedom in the introduc-
tion of new primitive concepts and new primitive sentences in the
language of physics or of science in general ; yet at the same time it
retains the possibility of differentiating pseudo-concepts and pseudo-
sentences from real scientific concepts and sentences, and thus of
eliminating the former, [This elimination, however, is not so
simple as it appeared to be on the basis of the earlier position of
the Vienna Circle, which was in essentials that of Wittgenstein.
On that view it was a question oi^^the language” in an absolute
sense ; it was thought possible to reject both concepts and sentences
if they did not fit into the language. ] A newly stated P-primitive
sentence is shown to be a pseudo-sentence if either no sufficient
rules of formation are given by means of which it can be seen to be
a sentence or no sufficient rules of transformation by means of
which it can, as previously indicated, be submitted to an empirical
test. The rules need not be explicitly given; they may also be
tacitly laid down, provided only that they are exhibited in the use
of language. A newly stated descriptive term is shown to be a
pseudo-concept if it is neither reduced to previous terms by means
of a definition, nor introduced by means of P-primitive sentences
that can be tested (see the example and counter-example on
p- 319)-
Like the individual sentences of the logic of science previously
discussed, this presentation of a conception of the logic of science
is intended only as an example. Its truth is not here in question.
The example is only for the purpose of making it clear that the
logical analysis of physics is the syntax of the physical language,
and of further stimulating the formulation, within the domain of
syntax, of views, questions, and investigations concerning the logic
of science (in the ordinary mode of expression : epistemology) and
thus making the subject more precise and more fruitful.
§ 83. The so-called Foundations of
THE Sciences
Much has been said in recent times about the problems of the
so-called philosophical or logical foundations of the individual
sciences, by which are understood (in our method of designation)
certain problems of the logic of science in relation to the domains
§83. THE SO-CALLED FOUNDATIONS OF THE SCIENCES 323
of the sciences. Taking the most important examples, we shall
show briefly that these problems are questions of the syntax of the
language of science.
The chief problems of the foundations of physics have already been
spoken of in the previous section, and, earlier, in Examples 49 to
53 (on p. 307). We have seen that the problem of the structure of
time and space is concerned with the syntax of the space and time
co-ordinates. The problem of causality is concerned with the syn-
tactical form of laws ; and in pcuticular the controversy regarding
determinism with a certain property of completeness of the system
of physical laws. The problem of empirical foundation (problem
of verification) is an inquiry into the form of the protocol-sentences
and the consequence-relations between the physical sentences —
especially the laws — and the protocol-sentences. The question of
the logical foundations of physical measurement is the question of
the syntactical form of quantitative physical sentences (containing
functors) and of the relations of derivation between these sen-
tences and the non-quantitative sentences (containing predicates;
for instance, sentences about pointer-coincidences). Further, such
questions as those concerning the relation between macro- and
micro-magnitudes or between macro- and micro-laws are to be
formulated as syntactical questions ; the elucidation of the concept
of genidentity also belongs to syntax.
The problems of the foundations of biology refer mainly to the con-
nection between biology and the physics of the inorganic, or, more
exactly, to the possibility of translating the biological language
into that sub-language S2 of the physical language which contains
the necessary terms for the purpose of describing the inorganic
processes and the necessary laws for the explanation of these pro-
cesses ; in other words : to the relations between and S2 on the
basis of the total language Sg which contains both as sub-languages.
There are, most importantly, two questions which must be dis-
tinguished: (i) Can the concepts of biology be reduced to those of
the physics of the inorganic? In syntactical form: Is every de-
scriptive primitive symbol of synonymous in S3 with a symbol
which is definable in 83 ? If this is the case, then there is in relation
to S3 an equipollent translation of the L-sub-language of S^ into
that of Sg. (2) Can the laws of biology be reduced to those of the
physics of the inorganic? In syntactical form: is every primitive
3H
PART V. PHILOSOPHY AND SYNTAX
law of Sj equipollent in S3 to a law which is valid in Sg ? If so, then
there is, in relation to S3, an equipollent translation of S^ (as a
P-language) into S2. This second question constitutes the scientific
core of the problem of vitalism^ which is, however, often entangled
with extra-scientific pseudo-problems.
The problems of the foundations of psychology contain analogues
to those of biology just mentioned, (i) Can the concepts of psycho-
logy be reduced to those of physics in the narrower sense?
(2) Can the laws of psychology be reduced to those of physics in
the narrower sense ? (Physicalism answers the first question in the
affirmative, but leaves the second open.) The so-called psycho-
physical problem is usually formulated as a question concerning the
relation of two object-domains : the domain of the psychical pro-
cesses and the domain of the parallel physical processes in the
central nervous system. But this formulation in the material mode
of speech leads into a morass of pseudo-problems (for instance:
“Are the parallel processes merely functionally correlated, or are
they connected by a causal relation ? Or is it the same process seen
from two different sides ?“). With the use of the formal mode of
speech it becomes clear that we are here concerned only with the
relation between two sub-languages, namely, the psychological and
the physical language; the question is whether two parallel sen-
tences are always, or only in certain cases, equipollent with one
another, and, if so, whether they are L- or P-equipollent. This im-
portant problem can only be grappled with at all if it is formulated
correctly, namely, as a syntactical problem — whether in the
manner indicated or in some other. In the controversy regarding
behaviorism there are two different kinds of question to be dis-
tinguished. The empirical questions which are answered by the
behavioristic investigators on the basis of their observations do
not belong here ; they are object-questions of a special science. On
the other hand, the fundamental question of behaviorism, which
is sometimes designated as a methodological or an epistemological
problem, is a problem of the logic of science. It is often formulated
in the material mode of speech as a pseudo-object-question (e.g.
“Do mental processes exist?”, “Is psychology concerned only
with physical behaviour?”, and so on). If, however, instead of
being formulated in this way it is formulated in the formal mode,
it will be seen that here again the question is one of the reducibility
§83. THE SO-CALLED FOUNDATIONS OF THE SCIENCES 325
of the psychological concepts; the fundamental thesis of be-
haviorism is thus closely allied to that of physicalism.
The problems of the foundations of sociology (in the widest sense,
including the science of history) are for the most part analogous to
those of biology and psychology,
§ 84. The Problem of the Foundation
OF Mathematics
What should a logical foundation of mathematics achieve? On this
question there are various views; the fundamental antithesis be-
tween them is particularly clearly brought out in two doctrines,
logicism^ which was founded by Frege (1884), and formalism, re-
presented by Frege’s opponents. (The designations ‘logicismj ;ind
‘formalism* only appeared later.) Frege’s opponents mcintained
that the logical foundation of mathematics is effected by the con-
struction of a formal system, a calculus, a system of axioms, which
makes possible the proof of the formulae of classical mathematics ;
in this the meaning of the symbols is not to be taken into con-
sideration, the symbols are, so to speak, implicitly defined by
the primitive sentences of the calculus; the question as to what
numbers actually are— -which goes beyond the domain of the
calculus — must be rejected. Formalism today represents a view
which is in essentials the same, but which has been improved upon
in several important points, notably by Hilbert. According to this
view, mathematics and logic are constructed together in a common
calculus ; the question of freedom from contradiction is made the
centre of the investigations; the formal treatment (the so-called
metamathematics) is carried out more strictly than before. As
opposed to the formalist standpoint, Frege maintained that the
logical foundation of mathematics has the task, not only of setting
up a calculus, but also, and pre-eminently, of giving an account of
the meaning of mathematical symbols and sentences. He tried to
perform this task by reducing the symbols of mathematics to the
symbols of logic by means .of definitions, and proving the sen-
tences of mathematics by means of the primitive sentences of
logic with the help of the logical rules of inference {\Grundgesetze]),
Later Russell and Whitehead, also representing the standpoint of
logicism, carried out in an improved form the construction of
336 PART V. PHILOSOPHY AND SYNTAX
mathematics on the basis of logic {{Princ, Ma/A.]). We will not go
into certain difficulties with which a structure of this kind is faced
(see Carnap \Logiziimus\)^ for we are here not so much concerned
with the question whether mathematics can be derived from logic
or must be constructed simultaneously with it, as with the question
whether the construction is to be of a purely formal nature, or
whether the meaning of the symbols must be determined. The
apparently complete antithesis of the opposing views on this point
can, however, be overcome. The formalist view is right in holding
that the construction of the system can be effected purely formally,
that is to say, without reference to the meaning of the symbols;
that it is sufficient to lay down rules of transformation, from which
the validity of certain sentences and the consequence relations be-
tween certain sentences follow; and that it is not necessary either
to ask or to answer any questions of a material nature which go
beyond the formal structure. But the task which is thus outlined
is certainly not fulfilled by the construction of a logico-mathe-
matical calculus alone. For this calculus does not contain all the
sentences which contain mathematical S3rmbols and which are
relevant for science, namely those sentences which are concerned
with the application of mathematics^ i.e. synthetic descriptive sen-
tences with mathematical symbols. For instance, the sentence
“In this room there are now two people present “ cannot be de-
rived from the sentence “ Charles and Peter are in this room now
and no one else ” with the help of the logico-mathematical calculus
alone, as it is usually constructed by the formalists; but it can be
derived with the help of the logicist system, namely on the basis of
Frege’s definition of ‘ 2 *. A logical foundation of mathematics is
only given when a system is built up which enables derivations of
this kind to be made. The system must contain general rules of
formation concerning the occurrence of the mathematical symbols
in synthetic descriptive sentences also, together with consequence-
rules for such sentences. Only in this way is the application of
mathematics, i.e. calculation with numbers of empirical objects
and with measures of empirical magnitudes, rendered possible and
systematized. A stnicture of this kind fulfils^ simultaneously, the
demands of both formalism and logicism. For, on the one hand, the
procedure is a purely formal one, and on the other, the meaning
of the mathematical symbols is established and thereby the appli-
§84. PROBLEM OF THE FOUNDATION OF MATHEJIATICS 327
cation of mathematics in actual science is made possible, namely,
by the inclusion of the mathematical calculus in the total language.
The logicist requirement only appears to be in contradiction with
the formalist one; this apparent antithesis arises as a result of the
ordinary formulation in the material mode of speech, namely, “an
interpretation for mathematics must be given in order that it may
be applied to reality”. By translation into the formal mode of
speech this relation is reversed: the interpretation of mathematics
is effected by means of the lulcs of application. The requirement of
logicism is then formulated in this way : the task of the logical foun-
dation of mathematics is not fulfilled by a metamathematics {that is,
by a syntax of mathematics) alone, but only by a syntax of the total
language, which contains both logico-mathematical and synthetic
sentences,
m
Whether, in the construction of a system of the kind described,
only logical symbols in the narrower sense are to be included
amongst the primitive symbols (as by both Frege and*Russell) or
also mathematical symbols (as by Hilbert), and whether only
logical primitive sentences in the narrower sense are to be taken as
L-primitive sentences, or also mathematical sentences, is not a
question of philosophical significance, but only one of technical
expedience. In the construction of Languages I and II we have
followed Hilbert and selected the second method. Incidentally,
the question is not even accurately formulated; we have in the
general syntax made a formal distinction between logical and
descriptive symbols, but a precise classification of the logical
symbols in our sense into logical symbols in the narrower sense
and mathematical symbols has so far not been given by anyone.
The logical analysis of geometry has shown that it is necessary
to distinguish clearly between mathematical and physical geo-
metry. The sentences belonging to the two domains, although they
often have the same wording in the ordinary use of language, have
a very different logical character. Mathematical geometry is a part
of pure mathematics, whether it is constructed as an a:domatic
system or in the form of analytical geometry. The questions of the
foundation of mathematical geometry thus belong to the syntax of
the geometrical axiom-systems, or to the syntax of the systems of
co-ordinates respectively. Physical geometry, on the other hand, is
a part of physics ; it arises from a system of mathematical geometry
328 PART V. PHILOSOPHY AND SYNTAX
by means of the construction of the so-called correlative definitions
(see § 25). In the case of the problems of the foundation of physical
geometry, the question is one of the syntax of the geometrical
system as a sub-language of the physical language. The principal
theses, for example, of the empiricist view of geometry: “The
theorems of mathematical geometry are analytic”, “The theorems
of physical geometry are synthetic but P-valid”, are obviously
syntactical sentences.
§ 85. Syntactical Sentences in the Literature
OF THE Special Sciences
In all scientific discussions, object-questions and questions of
the logic of science, i.e. syntactical questions, are bound up with
one another. Even in treatises which have not a so-called epi-
stemological problem or problem of foundation as their subject,
but are concerned with specialized scientific questions, a con-
siderable, perhaps even a preponderant, number of the sentences
are syntactical. They speak, for instance, about certain definitions,
about the sentences of the domain which have been hitherto
accepted, about the statements or derivations of an opponent,
about the compatibility or incompatibility of different assumptions,
and so on.
It is easy to realize that a mathematical treatise is predominantly
metamathematical, that is to say, that it contains, in addition to
proper mathematical sentences (for instance : “ Every even number
is the sum of two prime numbers”), syntactical sentences (of such
forms as: “From... it follows that...”, “By substitution we
get. . .”, “We will transform the expression. . and the like).
The same thing is equally true, however, of treatises of empirical
science. We will illustrate this by an example from physics. In the
following table the first column contains the initial sentences
(abbreviated) of Einstein’s Zur Elektrodynamik bewegter Korper
(1905). The reformulation in the second column is merely for the
purpose of making clear the character of the sentences. In the
third column, the character of the individual sentences or de-
scriptions is stated, and it is shown that the majority of these are
syntactical.
§ 85. SYNTACTICAL SENTENCES IN THE SPECIAL SCIENCES 329
Sentences from the
original
That Maxwell’s elec^
tro-dynamics ...
lead to asymmetries in
their application to
bodies in motion
which do not appear to
appertain to the phe-
nomena
is well known.
For example, if one
thinks of . . . reciprocal
causation ....
Here the observable
phenomenon is depen-
dent only upon the re-
lative motion of con-
ductor and magnet,
while, according to the
usual view, the case in
which the one body is
in motion must be
strictly separated from
the case in which the
other is in motion.
If, namely, the magnet
moves . . ., then an elec-
tric field ... is the re-
sult,
which proauces an elec-
tric current.
But if the magnet does
not move . . . then no
field ... results,
but on the other hand
an electro-motive power
results in the conduc-
tor ...»
Paraphrase
In the laws which are
consequences of the
Maxwell equations
certain asymmetries are
shown
which do not occur in
the appertaining proto-
col-sentences.
Contemporary physi-
cists know that ....
Example: the recipro-
cal causation-sentences
The protocol-sentences
are dependent only up-
on such and such sen-
tences of the system.
In the ordinary form of
the system the two
concrete sentences
and ‘ . . . * are not equi-
pollent to each other.
If a magnet moves...,
then an electric field ...
results.
If an electric field...
arises, a current ... re-
sults.
(Analogous.)
(Analogous.)
Kinds of sentence
(p.s. = pure-syntactical,
d.s. = descriptive -syn-
tactical.)
p.s. description of sen-
tences.
p.s. sentence about laws
and about protocol-sen-
tences.
Historical d.s. sentence.
r
p.s. description of sen-
tences.
t
p.s. sentence.
p.s. sentence (with de-
scriptions of two sen-
tences).
Object-sentence (phy-
sical law).
As before^
As before.
As before.
330
PARTY. PHILOSOPHY AND SYNTAX
Sentences from the
original
which, however, . . .
causes . . . electric cur-
rents.
Examples of a similar
kind,
like the unsuccessful
attempts to prove a
motion of the earth re-
lative to the “light
medium”,
lead to the supposition
thpt
...in electro-dynamics
no properties of the ob-
servable phenomena ...
correspond to the con-
cept of absolute rest,
but rather that ... the
same electro-dynamic
... laws are valid for all
co-ordinate systems ...
We will take this sup-
position
(whose content will be
called in what follows
the “ Principle of Rela-
tivity”)
as an hypo^esis.
Paraphrase
(Analogous.)
A I. Sentences similar
to the previous ones.
A 2 . Such and such
protocol-sentences oc-
curring in the history
of physics. By means
of these protocol-sen-
tences such and such an
hypothesis is refuted.
The sentences A sug-
gest the tentative con-
struction of a physical
system S for which the
sentences B are true
(that is to say, S is a
system of hypotheses
which is confirmed by
the sentences A).
Bi. There is no term
in the appertaining
protocol-sentences (of
the system S) corre-
sponding to the term
* absolute rest* in the
sentences of electro-
dynamics.
B 2. The . . . laws (of the
system S) have the
same form in relation
to all co-ordinate sys-
tems.
B2 shall be called the
“ Principle of Rela-
tivity’*.
B 2 is stated as a hypo-
thetical P-rule.
Kinds of sentence
As before.
(Loose) p.s. description
of sentences.
Historical d.s. descrip-
tion of sentences.
p.s. sentence
p.s. sentence.
p.s. sentence.
p.s. sentence (about
certain transforms -
tions).
p.s. definition.
p.s. convention (defini-
tion of * P-valid in S *).
§ 86. THE LOGIC OF SCIENCE IS SYNTAX
331
§ 86. The Logic of Science is Syntax
We have attempted to show by a brief examination of the pro-
blems of the logical analysis of physics and of the so-called pro-
blems of foundation ot the different domains — which also belong
to the logic of science — that these are, at bottom, syntactical,
although the ordinary formulation of the problems often disguises
their character. Metaphysical philosophy tries to go beyond the
empirical scientific questions of a domain of science and to ask ques-
tions concerning the nature of the objects of the domain. These
questions we hold to be pseudo-questions. The non-metaphysical
logic of science, also, takes a different point of view from that of
empirical science, not, however, because it assumes any mijta-
physical transcendency, but because it makes the language-forms
themselves the objects of a new investigation. On this view, it is
only possible, in any domain of science, to speak either in or about
the sentences of this domain, and thus only object-sentences and
syntactical sentences can be stated.
The fact that we differentiate these two kinds of sentences does
not mean that the two investigations must always be kept separate.
In the actual practice of scientific research, on the contrary, the
two points of view and the two kinds of sentences are linked with
one another. We have seen from the example of a treatise on physics
that investigations in the domains of the special sciences contain
many syntactical sentences. But it is also tnie, conversely, that
researches in the logic of science always contain numerous object-
sentences; these sentences are in part object-sentences of the
domain to which logical analysis is being applied, and in part
sentences concerning the psychological, sociological, and historical
circumstances under which work is being done in that field. So
although we can divide the concepts into logical and descriptive
concepts, and the sentences of simpler form into sentences of the
logic of science (that is to say, syntactical sentences) and object-
sentences, on the other hand no strict classification of the in-
vestigations themselves and the treatises in which they are set
forth is possible. Treatises in the domain of biology, for instance,
contain in part biological, and in part syntactical, sentences ; there
are only differences of degree, according to which of the two sorts
332
PART V. PHILOSOPHY AND SYNTAX
of question predominates ; and on this basis one may, in practice,
distinguish between specially biological treatises and treatises of
the logic of science. He who wishes to investigate the questions
of the logic of science must, therefore, renounce the proud claims
of a philosophy that sits enthroned above the special sciences,
and must realize that he is working in exactly the same held as the
scientific specialist, only with a somewhat different emphasis : his
attention is directed more to the logical, formal, syntactical con-
nections. Our thesis that the logic of science is syntax must there-
fore not be misunderstood to mean that the task of the logic of
science could be carried out independently of empirical science
and without regard to its empirical results. The syntactical in-
vestigation of a system which is already given is indeed a purely
mathematical task. But the language of science is not given to us
in a syntactically established form ; whoever desires to investigate
it must accordingly take into consideration the language which is
used in practice in the special sciences, and only lay down rules
on the basis of this. In principle, certainly, a proposed new syn-
tactical formulation of any particular point of the language of
science is a convention, i.e. a matter of free choice. But such a
convention can only be useful and productive in practice if it has
regard to the available empirical findings of scientific investigation.
[For instance, in physics the choice between deterministic laws
and laws of probability, or between Euclidean and non-Euclidean
geometry, although not univocally determined by empirical
material, is yet made in consideration of this material. ] All work
in the logic of science, all philosophical work, is bound to be un-
productive if it is not done in close co-operation with the special
sciences.
Perhaps we may say that the researches of non-metaphysical
philosophy, and especially those of the logic of science of the last
decades, have all, at bottom, been syntactical researches, although
unconsciously. This essential character of such investigations must
now also be recognized in theory and systematically observed in
practice. Only then will it be possible to replace traditional philo-
sophy by a strict scientific discipline, namely, that of the logic of
science as ihe syntax of the language of science. The step from the
morass of subjectivist philosophical problems on to the firm ground
of exact syntactical problems must be taken. Then only shall we
§ 86. THE LOGIC OF SCIENCE IS SYNTAX
333
have as our subject-matter exact terms and theses that can be
clearly apprehended. Then only will there be any possibility of
fruitful co-operative work on the part of the various investigators
working on the same problems — work fruitful for the individual
questions of the logic of science, for the scientific domain which is
being investigated, and for science as a whole. In this book we
have only created a first working-tool in the form of syntactical
terms. The use of this instrument for dealing with the numerous
and urgent contemporary problems of the logic of science, and the
improvement of it which will follow from its use, demands the co-
operation of many minds.
BIBLIOGRAPHY AND INDEX OF AUTHORS
The numbers immediately following an author’s name indicate
the pages of this book on which he is mentioned. The main refer-
ences are printed in black type.
The shortened forms of titles which precede them in square
brackets arc those used in citation throughout the book.
* The publications marked with an asterisk have appeared since
the writing of the German original, and hence are not mentioned
in the text. The most important of these are : Hilbert and Beniays
[GrundL 1934]; Quine [System\ (see the author’s review in
Erkenntnist 5, 1935, p. 285); Tarski [Wahrh,] (cf. Kokoszynska
[Wahrheit]).
Exhaustive bibliographies of the literature of logistics and logical
syntax are to be found in: Fraenkel [Mengenlehre]; Jorgensen
[Treatise\\ and Lewis [Survey^.
Ackermami, W.
Zum Hilbertschen Aufbau der reellen Zahlen. Math. Ann, 99,
1928.
Cber die ErfuUbarkeit gewisser Z^lausdriicke. Math. Ann, lOO,
1928.
See also Hilbert.
Ajdukiewicz, K., 167, 176, 227.
[Sprache] Sprache und Sinn. Erk, 4, 1934.
Das Weltbild und die Begriffsapparatur. Erk. 4, 1934.
*Die syntaktische Konnexitat. Studia Philos, i, 1935.
Ayer, A. J.
^Language, Truth and Logic. London, 1936.
Bachmann, F., see Carnap.
Becker, O., 46, 245, 246, 250, 254.
Mathematische Existenz. Jahrb. Phdnom. 1927; also published
separately.
[Modalitdten] Zur Logik der Modalit^ten. Jahrb , Phdnom. ii,
1930.
Behmann, H., 49 f., 139, 197, 246.
BeitrMge zur Algebra der Logik, insbesondere zum Entschei-
dungsproblem. Math. Ann. 86, 1922.
[Logik] Mathematik und Logik, Leipzig, 1927.
Entscheidungsproblem und Logik der Beziehungen. Jber. Math.
Vet. 36, 1928.
Zu den Widerspriichen der Logik. . . . Jber. Math. Ver. 40, 1931.
Sind die mathematischen Urteile analytisch oder synthetisch?
Erk. 4, 1934.
BIBLIOGRAPHY AND INDEX OF AUTHORS
335
Bemays, P., 96, 97, 173.
[Aussagenkalkul] Axiomatische Untersuchungen des Aussagen-
kalkiils der Priiicipia Mathematica. Math. ZS. 25, 1926.
With Schonfinkel: Zum Entscheidungsproblem der mathema-
tischen Logik. Math. Ann. 99, 1928.
[Philosophies Die Philosophic der Mathematik und die Hilhertsche
Beweistheonc. Bt. f. dt. Philos. 4, 1930.
See also Hilbert.
Black, M.
The Nature of Mathematics. London, 1933.
Blumberg, A. E. and Feigl, H.
Logical Positivism. Journ. of Philos. 28, 1931.
Borel, E.
Lemons sur la Thiorie des Fonctions. 3rd ed. Paris, 1928.
(Appendix: Discussion between R. Wavre and P. Iv^vy on
intiiitionist logic, reprinted from Revue M^taphys. 33, 19261,)
Br6al, M., 9.
Bridgman, P. W.
The Logic of Modem Physics. New York, 1927.
•A Physicist’s Second Reaction to Mengenlehre. Scripta Math.
2, 1934 (cf. Fraenkel [Diagonalverfahreti]).
Brouwer, L, E, J., 46 ff., 148, 161, 222* 269, 281, 305. (See also
Intuitionism.)
[Intuitionism\ Intuitionism and Formalism. Bull. Amer. Math.
Soc. 20, 1913-
Intuition istische Mengenlehre. Jber. Math. Ver. 28, 1920.
Intuitionistische Zerlegung mathematischer Grundbegriffe. Jber.
Math. Ver. 33, 1925-
t 5 ber die Bedeutung des Satzes vom ausgeschlossenen Dritten. . . .
Joum. Math. 154, 1925.
Intuitionistische Betrachtungen ubej* den Formalismus. Ber.
Akad. Berlin, Phys.-math. KL, 1928.
[SpracheS Mathematik, Wissenschaft und Sprache. Monatsh.
Math. Phys. 36, 1929.
Buhler, K., 9.
Cantor, G., 2671., 270.
Carnap, R.
[Aufbau] Der logische Aufbau der Welt. Berlin (now Meiner,
Leipzig), 1928.
[Scheinprobleme] Scheinprobleme m der Philosophic. Das Fremd-
psychische und der Realismusstreit. Berlin (now Leipzig), 1928.
[LogistikS Abriss der Logistik. (Schr. z. wiss. Weltauff.) Vienna,
1929.
Die alte und die neue Logik. Erk. i, 1930- (French transl.*
UAncienne et la Nouvelle Logique. Paris, 1933-)
336 BIBLIOGRAPHY AND INDEX OF AUTHORS
Camap, R.
[Axiomatikl Bericht iiber Untersuchungen zur allgemeinen
Axiomatik. Erk, I, 1930.
Die Mathematik als Zweig der Logik. BL /. du Philos, 4, 1930.
[Logizismus] Die logizistische Grundlegung der Mathematik.
Erk, 2, 1931.
[Metaphysik] Oberwindung der Metaphysik durch logische
Analyse der Sprache. Erk, 2, 1932. (French transl. : La Science
et la M^taphysique. Paris, 1934.)
[Phys, Sprache] Die physikalische Sprache als Universalsprache
der Wissenschaft. Erk, 2, 1932. (English transl. : The Unity of
Science, (Psyche Min.) London, 1934.)
[Psychol,] Psychologie in physikalischer Sprache. Mit Erwide-
rungen. Erk, 3, 1932.
[Protokollsdtze] tJber Protokollsatze. Erk, 3, 1932.
On the Character of Philosophic Problems. Phihs, of Science, i,
^ 1934*
Logische Syntax der Sprache, (Schr. z. wiss. WeltaufF.) Vienna,
1934. (The original of this book.)
Die Aufgabe der Wissenschaftslogik, (Einheitswiss.) Vienna,
1934. (French transl. : Le Problkme de la Logique de Science,
Paris, 1935O
[Antinomien] Die Antinomien und die Unvollstandigkeit der
Mathematik. Monatsh, Math, Phys, 41, 1934.
[GUltigkeitskriteriurn] Ein Gultigkeitskriterium fiir die Satze der
klassischen Mathematik. Monatsh, Math, Phys, 42, 1935.
* Philosophy and Logical Syntax, JPsyche Min.) London, 1935.
♦Formalwissenschaft und Realwissenschaft. Erk, 5, 1935.
*Les Concepts Psychologiques Rev. Synthese, 10, 1935.
With Bachmann, F., *tlber Extremalaxiome. Appearing in:
Erk, 6, 1936.
•Testability and Meaning. Appearing in: Philos, of Science,^, 1936.
Church, A., 160.
A Set of Postulates for the Foundation of Logic. Ann. of Math,
33, 1932; 34, 1933-
•The Richard Paradox. Amer, Math. Monthly, 41, 1934.
•An Unsolvable Problem of Elementary Number Theory. Jimer,
Joum, Math. 58, 1936.
Chwistek, L., 9, 213, 245, 246, 249.
Uber die Antinomien der Prinzipien der Mathematik. Math. ZS.
14, 1922.
Sur les Fondements de la Logique Moderne. Atti V. Congr,
Intern, Filos, (1924), 1925.
Neue Grundlagen der Logik und Mathematik. I, Math. ZS, 30,
1929 ; II, 34» 1932-
[Nom. Grundl.] Die nominalistische Grundlegung der Mathe-
matik. Erk, 3, 1933.
BIBLIOGRAPHY AND INDEX OF AUTHORS
337
Chwistek, L.
With W. Hetper and J. Herzberg: Fondements de la M^ta-
math^atique rationelle. BtdLAcad, Pol., S^. A: Math., 1933.
As above : Remarques sur la . . . M^tamath^matique rationelle.
Loc. cit.
Curry, H. B. ,
An Analysis of Logical Substitution. Amer.Joum. Math. 51, 1929.
Grundlagen der kombinatorischen Logik. Amer. Journ. Math.
52, 1930.
Apparent Variables from the Standpoint of Combinatory Logic.
Ann. of Math. 34, 1933.
♦Functionality in Combinatory Logic. Proc. Nat. Acad. Sci. 20,
1934 -
Dedekind, R., 137.
Was sind und was sollen die Zahlen? Brunswick, 1888.
Dubislav, W., 44.
[Analyt.^ Ober die sog. analytischen und synthetischen Urteile.
Berlin, 1926.
Zur kalkiilmMssigen Charakterisierung der Definitfonen. Ann.
Philos. 7, 1928.
Elementarer Nachweis der Widerspruchslosigkeit des Logik-
kalkiils. Joum. Math. 161, 1929.
Die Definition. Leipzig, 3rd ed., 1931.
Die Philosophie der Mathematik in der Gegenwart. Berlm, 1932.
Naturphilosophie, Berlin, 1933.
Duhem, P., 318.
Diirr, E.
[Leihniz\ Neue Beleuchtung einer Theorie von Leibniz. Darmstadt,
1930.
Einstein, A., 178, 328.
Feigl, H.
♦The Logical Character of the Principle of Induction. Philos, of
Sci. I, 1934.
♦Logical Analysis of the Psycho-Physical Problem. Philos, of Sci.
I, 1934 -
See also Blumberg.
Fraenkel, A., 97 f., 162, 213, 267 ff., 270, 274, 275, 335 -
[Untersuchungeri] Untersuchungen iiber die Grundlagen der
Mengenlehre. Math. ZS. 22, 1925.
Zehn Vorlesungen fiber die Grundlegung der Mengenlehre. Leipzig,
1927.
{Mengenlehre^ Einleitung in die Mengenlehre. Berlin, 3rd ed., 1928.
Das Problem des Unendlichen in der neueren Mathematik.
Bl.f. dt. Philos. 4, 1930.
SL
23
BIBLIOGRAPHY AND INDEX OF AUTHORS
338
Fraenkel, A.
Die heutigen GegensStze in der Gnmdlegung der Mathematik.
Erk, I, 1930.
*Sur la Notion d’Existence dans les Math^matiques. Enseign.
Math. 34, 1935.
•Sur l*Axiome du Choix. Loc. cit.
*[Diag(malveffahren] Zum Diagonalverffihren Cantors. Fund.
Math. 25, 1935.
Frank, Ph., 280 f.
Was bedeuten die gegenwSrtigen physikalischen Theorien fiir die
allgemeine Erkenntnislehre? Natunviss. 17, 1929; also in: Erk.
I, 1930.
[Kausalgesetz] Das Kausalgesetz und seine Grenzen. (Schr. z. wiss.
Weltauff.) Vienna, 1932.
Frege, G., 44, 49, 99, 134, 136 ff., 143, 144, 158, 197, 203, 259,
,395. 335 ff-
BegrifFsschrift. Halle, 1879.
[Grundlagen]Die Grundlagender Arithmetik. Breslau, 1884. (New
ed. 1934.)
[Grundgesetze] Grundgesetze der Arithmetik. Jena, I, 1893; II,
1903.
[Zahleri\ Vher die Zahlen des Herm H. Schubert. Jena, 1899.
Gi&tschenberger, R.
Sytnbola. AnfangsgrUnde einer Erkermtnistheorie. Karlsruhe, 1920.
Zeichen^ die Fundamente des Wissens. Eine Absage an die Philo-
Sophie. Stuttgart, 1932.
Gentzen, G.
*Die Widerspruchsfreiheit der reinen Zahlentheorie. Math.
Ann. 112, 1936.
*Die Widerspruchsfreiheit der Stufenlogik. Math. ZS. 41, 1936.
Glivenko, V., 227*
G6del, K., 28, 55, 96 f., 99, 100, 106 f., 129, 130, 131 ff., 134, 139,
160, 173, 189, 197, 209, 211, 219, 227, 250.
Die Vollst^digkeit der Axiome des logischen Funktionenkalkiils.
Monatsh. Math. Phys. 1930.
[Unentscheidbare] tJber formal unentscheidbare Satze der Prin-
cipia Mathematica und verwandter Systeme. I. Monatsh.
Math. Phys. 38, 1931.
[Kolloquium] Various notes in: Ergebn. e. math. Kolloquiums
(K. Menger). Vols. 1-4, i93i“33'
Grelling, K. and Nelson, L., 211 fl.
Bemerkungen zu den Paradoxien von Russell und Burali-Forti.
Abh. d. Friesschen Schtde, N.F. 2, 1908.
BIBLIOGRAPHY AND INDEX OF AUTHORS
339
Hahn, H., 280.
\Wiss. Weltauff.\ Die fiedeutung der Avissenschaftlichen Welt-
auffassung, insbes. for Mathematik und Physik. Erk, i, 1930.
Logik, Mathematik und Naturerkennen. (Einheitswiss.) Vienna,
1933 -
Helmer, O. •
^Axiomatischer Aufbau der Geometrie in formalisierter Darstellung.
Diss., Berlin, 1935.
Hempel, C. G., 317.
*Beitrdge zur logischen Analyse des Wahrscheinlichkeitsbegriffes,
Diss., Berlin, 1934.
•Analyse Logique de la Psychologie. Rev, Synthkse, lO, 1935.
•Ober den Gehalt von Wahrscheinlichkeitsaussagen. Erk. 5,
1935 -
With P. Oppenheim. *Der Typusbegriff im Lichte der neuen
Logik, Leyden, 1936. ^
Herbrand, J., 53, 134, 173.
Recherches sur la Thdorie de la D^onstration, Th^se Fac. Sciences
Paris (Nr. 2121, s^rie A, 1252), 1930. Also in; Travaux Soc,
Sciences Varsovie, Cl. iii, Nr. 33, 1930.
[Non~Contrad,] Sur la Non-Contradiction de rArithm^tique.
Joum, Math, 166, 1931.
Sur le Probl^me fundamental de la Logique mathematique.
C,R, Soc, Sciences Varsovie, 24, Cl. iii, 1931.
Hertz, P., 275.
[Axiom,] Cber Axiomensysteme fur beliebige Satzsysteme. Math.
Ann. loi, 1929.
Vom Wesen des Logischen.. . . Erk. 2, 1932.
Herzberg, J. See Chwistek.
Hetper, W.
•Semantische Arithmetik. C,R. Soc. Sciences Varsovie, 27, Cl. iii.
1934,
See also Chwistek.
Heyting, A., 46 ff., 166, 203, 222, 227, 245, 246, 249 f-
[Logik] Die formalen Regeln der intuitionistischen Logik. Ber,
Akad, Berlin, 1930.
[Math.] Die formalen Regeln der intuitionistischen Mathematik.
I, II. Loc. cit.
[Grundlegung] Die intuitionistische Grundlegung der Mathe-
matik. Erk. 2, 1931.
Anwendung der intuitionistischen Logik auf die Definition der
Vollstandigkeit eines Kalkuls. Intern. Math.-Kongr. Zurich,
1932.
*Mathematische Grundlagenforschung, Intuitionismus, Beweis-
theorie. (Erg. d. Math., Ill, 4.) Berlin, 1934.
340 BIBLIOGRAPHY AND INDEX OF AUTHORS
Hilbert, D., 9, 12, 19, 35» 36, 44 48, 49, 79, 97, 99, 104, 128 f.,
140, 147, 158, 160, 173, 189, 197, 203, 244, 259, 274,
281, 325, 327.
[GrundL Geom,] Grundlagen der Geomettie, Leipzig, 1899.
7th eds 1930.
Axiomatisches Denken. Math. Ann. 78, 1918.
NeubegrQndimg der Mathematik. Ahh. Math. Sent. Hamburg, i,
1922.
[Grtmdl. 1923] Die logischen Grundlagen der Mathematik.
Math. Ann. 88, 1923.
[Unendliche] t)ber das Unendliche. Math. Ann. 95, 1926.
Die Grundlagen der Mathematik. Mit Bemerkungen von Weyl
und Bemays. Ahh. Math. Sent. Hamburg, 6, 1928.
With Ackermann: [Logik\ GrundzUge der theoretischen Logik.
Berlin, 1928.
Probleme der Grundlegung der Mathematik. Math. Ann. 102,
.1930.
[Grundl. 1931] Grundlegung der elementaren Zahlenlehre.
Math. Ann. 104, 1931.
[TertiuM\ BeWeis des Tertium non datur. Nachr. Ges. Wiss.
Gdttingen, math.-phys. Kl., 1931.
With Bemays: ^[Grundl. 1934] Grundlagen der Mathematik. I.
Berlin, 1934.
Hume, D., 280.
Huntington, E. V.
Sets of Independent Postulates for the Algebra of Logic. Trans.
Amer. Math. Soc. 5, 1904.
A New Set of Postulates for Betweenness, with Proof of Complete
Independence. Trans. Amer. Math. Soc. 26, 1924.
A New Set of Independent Postulates for the Algebra of Logic,
with Special Reference to Whitehead and Russeirs Principia
Mathematica. Proc. Nat. Acad. Sci. 18, 1932-
Husserl, E., 49.
Jaskowski, St.
•On the Rules of Suppositions in Formal Logic. (Studia Logica,
Nr. I.) Warsaw, 1934.
jargensen, J., 258, 335.
\TTeatise\ A Treatise of Formal Logic. Its Evolution and Main
Branches toith its Relation to Mathematics and Philosophy.
3 vols. Copenhagen, 1931.
[ZtWe] Ober die Ziele und Probleme der Logistik. Erk. 3, 1932.
Kaufmann, Felix, 46, 51 f., 139, 161, 165.
[Unendliche] Das Unendliche in der Mathematik und seine Aus~
schaltung. Vienna, 1930.
[Bemerkungen] Bemerkungen zum Gnindlagenstreit in Logik und
Mathematik. Erk. 2, 1931.
BIBLIOGRAPHY AND INDEX OF AUTHORS
341
Kleene, S. C.
•A Theory of Positive Integers in Formal Logic. Amer, Joum,
Math. 57, 1935.
With Rosser, J. B. •The Inconsistency of Certain Formal
Logics. Ann. of Math. 36, 1935.
Kokoszynska, M.
*\Wahrheii\ t^ber den absoluten Wahrheitsbegriff und einige
andere semantische Begriffe. Erk. 6, 1936.
Kronecker, L., 305.
Kuratowski, C. See Tarski.
Langford, C. H. See Lewis.
Leibniz, 49.
Le^niewski, St, 24, 160.
[Neues System\ Grundzuge eines neuen Systems der Grundlagen
der Mathematik. Fund. Math. 14, 1929.
[Ontologie\ Uber die Grundlagen der Ontologie. C.R. Soc,
Sciences Varsovie, 23, Cl. iii, 1930.
[Definitionen] t)ber Definitionen in der sog. Theorie der Deduk-
tion. Ibid. 24, Cl. ill, 1931. •
L6vy, P. See Borel.
Lewis, C. I., 12, 203, 223 f., 232, 245, 246, 250, 252, 253 f., 257 f.,
275,281,335-
[Survey\ A Survey of Symbolic Logic. Berkeley, 1918,
Alternative Systems of Logic. Monisty 42, 1932.
With Langford, C. H. : \Logic\ Symbolic Logic. New York and
London, 1932.
•Experience and Meaning. Philos. Review y 43, 1934.
Lowenheim, 270.
Lukasiewicz, J., 9, 96, 160, 250, 254.
With Tarski: [Aussagenkalkul] Unteisuchungen iiber den Aus-
sagenkalkiil. C.R. Soc. Sciences Varsoviey 23, Cl. iii, 1930-
\Mehrwertige\ Philosophische Bemerkungen zu mehrwertigen
Systemen des Aussagenkalkiils. Loc, cit.
MacColl, 254.
MacLane, S.
*Abgekurzte Beweise zum Logikkalkul. Diss. Gottingen, 1934-
Mannoury, G.
•Die signifischen Grundlagen der Mathematik. Erk. 4, 1934.
Menger, K., 52.
Bemerkungen zu Grundlagenfragen (especially II: Die mengen-
theoretischen Paradoxien). Jber. Math. Vet. 37, 1928.
[Intuitionismus^ Der Intuitionismus. Bl.f. dt. Philos, 4, 193^-
Die neue Logik. In : Krise und Neuaufbau in den exakten Wissen-
schaften. Vienna, 193 3-
342
BIBLIOGRAPHY AND INDEX OF AUTHORS
Mises, R. V., 149.
Wahrscheinlichkeitt Statistik und Wakrheit. (Schr. z. wiss.
Weltauff.) Vienna, 1928.
Ober das naturwissenschaftliche Weltbild der Gegenwart.
Natunmss. 19, 1931.
Morris, C. W.
The Relation of Formal to Instrumental' Logic. In : Essays in
Philosophy ^ 1929.
*The Concept of Meaning in Pragmatism and Logical Positivism.
Proc, Intern, Congr, Philos. (1934). Prague, 1936.
Nagel, E.
^Impressions and Appraisals of Analytic Philosophy in Europe.
Joum. of Philos. 33, 1936.
Nelson, E. J., 254, 257.
[Intensional] Intensional Relations. Mindy 39, 1930.
Deductive Systems and the Absoluteness of Logic, Mindy 42,
•T933-
On Three Logical Principles in Intension. Monist, 43, i933-
Neumann, J. v., 96, 98, 139, 147, 166, 173.
[Bev 3 eisth.'\ Zur Hilbertschen Beweistheorie. Math. ZS. 26, 1927.
Die formalistische Grundlegung der Mathematik. Erk. 2, 1931*
Neurath, O., 280, 281, 283, 286, 320!.
With others: [Wiss. Weltauff. Wissenschaftliche Weltauff assung.
Der Wiener Kreis. (Verdff. d. Vereins Ernst Mach.) Vienna,
1929.
[Wege'\ Wege der wisspnschaftlichen Weltauffassung. Erk. l,
1930.,
Empirische Soziologie. Der wissenschaftliche Gehalt der Geschichte
und NationalSkonomie. (Schr. z. wiss. Weltauff.) Vienna, 1931.
[Physicalism\ Physicalism. The Philosophy of the Viennese
Circle. Menisty 41, 1931.
[Physikalismus'] Physikalismus. Scientiay 50, 1931.
[Soziol. Phys.] Soziologie im Physikalismus. Erk. 2, 1931.
[Protokollsdtze] ProtokollsUtze. Erk. 3, 1932.
[Psychol.'] Einheitswissenschaft und Psychologic. (Einheitswiss.)
Vienna, 1933-
*Radikaler Physikalismus und “wirkliche Welt”. Erk. 4, i935*
•Le Diveloppement du Cercle de Vienne et V Avenir de V Empiricisme
logique. Paris, 1935.
Nicod, J.
A Reduction in the Number of the Primitive Propositions of
Logic. Proc. Cambr. Phil. Soc. 19, 1917.
Ogden, C. K. and Richards, 1 . A.
The Meaning of Meaning. A Study of the Influence of Language
upon Thought and of the Science of Symbolism. London, 1930.
Oppenheim, P. Sec Hempel.
BIBLIOGRAPHY AND INDEX OF AUTHORS
343
Parry, W. T., 254, 257*
\KolL'\ Notes in: Erg, e, math. Kolloqtdums (ed. by Menger).
Heft 4, 1933.
Peano, G., 31 f., 44, 97, 99, 144, 158, 166, 212.
Notations de Logique mathAnatique. Turin, 1894.
[Formulatre] Formulaire de MathAnatiques. Turin (1895), 1908.
Peirce, Ch. S.
Collected Papers. Ed. by Ch. Hartshome and P. Weiss. 5 vols.
Cambridge, Mass. 1931 fF. (Especially Vols. 2-4.)
Penttild, A. and Saamio, U.
•Einige grundlegende Tatsachen der Worttheorie. . . . Erk. 4,
1934 -
Poincar^, H., 46, 16 1, 269, 318.
Wissenschuft und Hypothese. Leipzig (1904), 1914.
Wissenschaft und Methode, Leipzig, 1914.
[Gedanken] Letzte Gedanken. Leipzig, 1913. ^
Popper, K., 317, 321.
^Logik der Forschung. Zur Erkenntnistheorie der modemen Natur-
wissenschaft. (Schr. z. wiss. WeltaufF.) Vienna, f935.
Post, E. L., 208.
[Introduction] Introduction to a General Theory of Elementary
Propositions. Amer. Joum. Math. 43, 1921.
Presburger, M.
C ber die Vollstandigkeit eines gewissen Systems der Arithmetik. . . .
Congr. Math. Warschau (1929), 1930.
Quine, W. V., 190.
^[System] A System of Logistic, Cambridge, Mass., 1934.
•Ontological Remarks on the Propositional Calculus. Mindy 43,
1934 -
•Towards a Calculus of Concepts. Joum. Symbol, Logicy I, 1936.
•Truth by Convention. In: Philosophical Essays for A. N,
Whiteheady edited by O. H. Lee, 1936.
•A Theory of Classes Presupposing No Canons of Type. Proc,
Nat, Acad, Sci. 22, 1936.
•Definition of Substitution. Bull, Amer. Math. Soc. 42, 1936.
•Set-Theoretic Foundations for Logic. Journ. Symbol. LogiCy I,
1936.
Ramsey, F. P., 50, 86, 114, 211 f., 213, 283. 3^1.
[Foundations] The Foundations of MathematicSy and Other Logical
Essays. London, 1931-
Reichenbach, H., 78, 281, 31 1, 3 I 7 -
[Axiomatik] Axiomatik der relativistischen Raum~Zeit~Lehre.
Brunswick, 1924.
[Philosophic] Philosophic der Raum-Zeit-Lehre. Berlin, 1928.
344
BIBLIOGRAPHY AND INDEX OF AUTHORS
Reichenbach, H.
[Wahrscheinlichkeitslogik] Wahrscheinlichkeitslogik. Ber. Akad.
Berlin, 29, 1932.
^Wahrscheinlichkeitslehre, Eine Untersuchung tiber die logischen
und rnathematischen Grundlagen der Wahrscheinlichkeitsrechnung.
Leyden, 1935.
Richard, J., 213, 219, 222, 270.
Richards, I. A. See Ogden.
Rosser, J. B.
*A Mathematical Logic without Variables. I. Ann. of Math. 36,
1935. 11. Duke Math. Joum. l, 1935.
See also Kleene.
Rtistow, A., 213.
Der LUgner. Theorie, Geschichte und Aufldsung. Diss. Erlangen,
1910.
Russell, B., 19, 23, 35, 44 f., 47 ff., 49 ff., 86, 96 f., 99, 134, 136 ff.,
140, 143, 144 f-. *58, 160, 162, 164 f., 173, 189, 192, I9S, 197,
203, 2ii f., 231, 244, 245 f., 249, 253-i5S, 257 f-. 259 f., 261,
281, 291, 293, 395. 325 f-. 327-
[Principles] The Principles of Mathematics. Cambridge, 1903.
The Theory of Implication. Amer. Journ. Math, 28, 1906.
With Whitehead: [Princ. Math.] Principia Mathematica, I (1910),
1925; 11 (1912), 1927; III (i9i3)» 1927-
Our Knowledge of the External World. New York, 1914.
[Math. Phil.] Introduction to Mathematical Philosophy. 1919.
[Introd. Wittg.] 1922. See Wittgenstein.
Schlick, M., 51, loi, 280 f., 284, 310, 321.
Allgemeine Erkermtnislehre. Berlin (1918), 2nd ed. 1925.
[Metaphysik] Erleben, Erkennen, Metaphysik. Kantstud.^l, 1926.
[Wende] Die Wende der Philosophic. Erk. i, 1930.
[Kausalitdt] Die KausalitSt in der gegenwartigen Physik. Natur-
toiss. 19, 1931.
[Positivismus] Positivismus und Realismus. Erk. 3, 1932.
^[Fundament] t)ber das Fundament der Erkenntnis. Erk. 4, 1934.
•Meaning and Verification. Philos. Review, 45, 1936.
Scholz, H., 258, 260.
[Geschichte] Geschichte der Logik. Berlin, 1931.
With Schweitzer, H. : *Die sog. Definitionen durch Abstraktion.
(Forsch. z. Logistik, No. 3.) Leipzig, 1935.
Schdnfinkel, M.
Ober die Bausteine der rnathematischen Logik. Math. Ann. 92,
1924.
See also Bemays.
BIBLIOGRAPHY AND INDEX OF AUTHORS 345
Schr6der, E., 44, 158.
Vorlesungen iiber die Algebra der Logik {exakte Logik). 3 vols.
Leipzig, 1890-1905.
Sheffer, H. M.
A Set of Five Independent Postulates for Boolean Algebras.
Trans, Amer, Math, Soc, 14, 1913.
Mutually Prime Postulates. Bull, Amer, Math, Soc, 22, 1916.
Skolem, Th., 270.
[Erfiillbarkeii] Logisch-kombinatorische Untersuchungen iiber
die Erfiillbarkeit oder Be'veisbarkeit mathematischer S^tze.
Vidensk, Skr, Kristiania, 1920, No. 4.
Begriindung der elementaren Arithmetik durch die rekurrierende
Denkweise. Vidensk. Skr, Kristiania, 1923, No. 6.
Cber einige Grundlagenfragen der Mathematik. Skr. Norske
Vid.^Akad, Oslo. I. Mat. Nat. Kl. 1929, No. 4.
Tarski, A., 32, 70, 89, 96 f., 160, 167, 172, 173, i97» 204,^08,
209, 275.
Sur le Terme Primitive de la Logistique. Fund. Math. 4, 1923.
Sur les Truth-Functions au Sens de MM. Whitehead and Russell.
Fund. Math. 5, 1924.
Ober einige fundamentale Begriffe der Metamathematik. C,R.
Soc. Sciences Varsovie, 23, Cl. iii, 1930.
[Methodologies Fundamentale Begriffe der Methodologie der de^
duktiven Wissenschaften. I. Monatsh. Math. Phys. 37, 1930*
Sur les Ensembles d^finissables de Nombres r^els. I. Fund.
Math. 17, 1931-
With Kuratowski, C. : Les Operations logiques et les Ensembles
projectifs. Fund. Math. I7» i93i*
[WahrheitsbegriffS Der Wahrheitsbegriff in den Sprachen der
deduktiven Disziplinen. Anzeiger Akad. Wien, 1932, No. 2.
(Note on a Polish treatise; German translation: [Wahrh.S)
[Widerspruchsfr.] Einige Betrachtungen uber die Begriffe der
o>-Widerspruchsfreiheit und der cu-Vollstandigkeit. Monatsh.
Math. Phys. 40, 1933.
•Einige methodologische Untersuchungen uber die Definier-
barkeit der Begriffe. Erk. 5, 1935-
♦Grundziige des Systemenkalkiils. I. Fund. Math. 25, 1935*
*[Wahrh.S Der Wahrheitsbegriff in den formalisierten Sprachen.
Stud. Philos. I, 1936.
See also Lukasiewicz.
Vienna Circle, 7, 44» 280, 282, 309> 3^^ Carnap, FeigI,
Frank, G6del, Hahn, Neurath, Schlick, Waismann.
Waismann, F., 321.
Die Natur des Reduzibilitatsaxioms. Monatsh. Math. Phys. 35,
19284
BIBLIOGRAPHY AND INDEX OF AUTHORS
346
Waismann, F.
[Wahrscheinlichkeiil Logische Analyse des Wahrscheinlichkeits-
begriffes. Erk, 1, 1930.
^Ober den Begriff der IdentitHt. Erk, 6 , 1936.
Wajsberg, M.
Uber Axiomensysteme des AussagenkalkUls. Monatsh. Math.
Phys. 39, 1932.
Ein erweiterter KlassenkalkiU. Monatsh, Math, Phys, 40, 1933.
Untersuchungen iiber den Funktionenkalkiil fUr endliche Indi-
viduenbereiche. Math, Ann, 108, 1933.
Beitrag zur Metamathematik. Math, Atm, 109, 1933.
^BeitrSge zum MetaaussagenkalkUl. Monatsh, Math, Phys, 41,
1934.
Warsaw logicians, 9, 160, 281. See also Le^niewski« Lukasiewicz,
Tarski.
Wavre, R. See Borel.
Weiss, P., 258.
The Nature of Systems, (Reprinted from Monist,) 1928.
Two-Va?.ued Logic, another Approach. Erk, 2, 1931.
Weyl, H., 46, 99, 148, 186, 305.
\Kontinuum\ Das Kontinuum, Leipzig, 1918.
Ober die neuere Grundlagenkrisc der Mathematik. Math, ZS,
10, 1921.
Die heutige Erkenntnislage in der Mathematik. Sympos, I, 1925.
(Also published separately.)
Philosophie der Mathematik und Naturwissenschaft. Part I.
In: Handb, d, Philos. ^ ed. by B^umler and Schroter, Munich,
1926. (Also published separately.)
Whitehead, A. N., 44, 99, 158.
Metaphysics and Logic of Classes. Monist ^ 42, 1932.
See also Russell.
Wittgenstein, L., 44, 46, 49 ff., 51 f., 53, loi, 139, 140, 161, 186,
199, 245 f., 280, 282 ff., 295 f*. 303, 304, 307, 321 f.
[Tractatus'] Tractatus Logico-Philosophicus, With introd. by
B. Russell. London, 1922.
Zermelo, E., 93, 97.
Untersuchungen ttber die Grundlagen der Mengenlehre. Math,
Ann, 65, 1908.
INDEX OF SUBJECTS
The numbers refer to pages. The most important passives are indicated
by black type.
Abbreviations : I = Syntax of Language I
II = Syntax of Language II
. G = General Syntax
see Expression
Q, see Symbol
.Abbreviation, 157 f.
Accent, Accented expression, 1 : 13,
26; II: 132
?lfu, see Expressional function
?Ig, see Expressional framework
Aggregate, see Class
Aggregates, Theory of, II: 83, 86,
97 f., 138; G: 221, 267
Analytic, 1 : 28, 39 f., 43 ; II : 100 f..
Ill f., 124, 132 f. ; G: 182 fT.
Antinomy, 3, 137 f., 21 iff., 2171!.,
221
Argument, Argument-ex-
pression, I: 26; II: 81, 87 f.;
G: 187 ff.
Based, 63, 68, 72
Behaviourism, 324
C-, II: 100 f.; G: 171, 172 ff.,
175, 182, 183, 185
Calculable, 148 f.
Calculus, 4 ff., 167, 228 f.
Characteristic, G : 202
Class, 37, 97, 134 ff., 136 ff.
Class of expressions, 1 : 37 ; G : 169
Closed, I: 21, 66; G: 194
Coextensive, 137; G: 187 f., 241
Compatible, 1 : 40 ; II : ii7;G: 174
Complete, i. c. 6, it, G: 175, 199
2. c. language, G : 209
Conjunction, I: 18 f.; II: 89, 103;
G: 202
Conjunctive standard form, 103
Consequence, 27; I: 37, 38 f.;
II: iiyff.; G: 168, 172, 254
Consequence, Rules of, I: 37 ff.;
II: 98 ff.; G: 171
Consequence-class, I ; 38 ; G : 172
Consequence-series, I: 38 f.; G:
172
Argument, Suitable, G: 188
Arithmetic, I: 30 f., 59; II: 97,
134; G: 169, 205 ff., 220 ff.,
304f.»325ff-
Arithmetization, 54 ff., 57, 79
Atomic sentence, II : 88
Autonymous, 17, 153, 156!., 160,
237, 238, 247 f.
Autonymous mode of speech, see
Mode ^
Axiom, see Primitive sentence.
Principle
Axiom-system, G : 274 f.
Axiomatic method, system, 76,
78 ff., 271 ff.
Axiomatics, 274 f.
Bound (d), 1 : 21, 66 ; II : 87 ; G : 192
Bracket, 15, 19
Consistent, G : 207, 275
Constant, I: 16; II: 84; G: 194
Content, I: 42; II: 120; G: 175 f.
Continuum, 305
Contra-class, -sentence, G : 200, 203
Contradiction, 137, 291, 297, 304;
see also Antinomy, and Prin-
ciple of
Contradictory, I: 28, 39 ff., 44;
II: iiif., 128; G: 182, 207,
275
Contravalid, G: 174, 275
Convaluable, II: 108
Converse, 264
Co-ordinate, 12
Co-ordinates, Language of, 12, 45
Correlate, 222
Correlated syntactical sentence,
234. 23*
Correlation, Syntactical, 222
Correlative definition, 7, 78, 79
Correlator, G : 265
348
INDEX OF SUBJECTS
D
b, see Descriptive
d-, II: 99, loi; G: 170 ff., 174 f*.
182, 183, 185
Defuiable, 114
Defined, G: 172
Definiendum, definiens, 23
Definite, 11; I: 45 f.; II: 98,
160 ff., 165 ff., 172; G: 198
Definition, I: 23 f., 37, 66 ff., 72,
78; II: 88 f.; G: 172, 194
Definition, Explicit, 23, 24 ; II : 88 f.
Regressive, 23, 68; II: 88, 89
Detinition-chain, 24, 71
Definition in usu, 24
Definition-schema, 68
Definition-sentence, II: 71, 72
Demonstrable, 1 : 28 f., 75 f. ;
II: 94, 124: G: 171
DenVmerable, 213, 220, 268 ff.
Dependent, G: 174
Derivable, I: 27 f., 39, 75 i 94;
G: 171'
Derivation, I: 27!., 33 ff., 39, 75;
II: 94 f.; G: 171
Description, descriptions!, 1 : 22 f.,
144 ff.. 154 ff.; G: 193, 195,
291
Descriptive, i. d. I: 13, 14, 25,
38; II: 72; G: 177 f., 230,
231 f.
2. d. Language, G: 178, 210
3. d. Syntax, 7, 53, 56 ff., 79 f.,
131, 154
Design (Gestalt)^ 15 f., 91, 155
Designation and designated, 18,
153-160
Designation, Syntactical, 154!., 160
Determinate, i. d. 0, R, II: loi,
ii5;G: 174
2. d. Language, G: 209
Diagonal method, 221
Direct consequence, I: 38; G:
168 ff., 170 f.
Directly derivable, 27; I: 32, 74;
II: 94, G: 171
Disjunction, I ; 19 f. ; II 89, 103 ;
G : 202 f .
Double negation, I: 34; II: 125
E
Elementary sentence, 166; G: 195
Elimination, 1 : 24 f., 31 ; II : 89 f. ;
G: 172
“Elucidation**, 283, 321
Empty, GT: 261
Equality, see Identity
Equation, I: 19, 36; G: 243
Equipollent, I: 42; II: 120; G:
176, 184, :^4I
Equipollent translation, G : 226
Equivalence, I: 19; G: 202
Equivalence, Symbol of, I: 16, 19,
49; II: 84; G: 243
Evaluation, II: 108, no
F
f , see Functor-variable
False, 214, 217
Form, 16, 155
Formal, 1, 258 f., 281 f.
Formal mode of speech, see Mode
Formalism, 300, 325 ff.
Formation rules, 2,4;!: 26, 62 ff. ;
II: 87ff.;G: 167 ff.
Free (o), I: 21, 66; II: 87; G: 192
see Functor-expression
fu, see Functor
Excluded Middle, see Principle of
Exclusive content, G: 176
Existence, 140 f.
Existential operator, I: 21 ; G: 93,
196 f.
sentence. Unlimited, 47, 163
Expression, 4; I: 16; G: 167 f.
Expression, Principal, G: 177
Expressional framework, G: 187
faction, G: 191
Extensional, 93, 240 ff.
Extensionality, Primitive sentence.
Axiom, of, see Principle
Thesis of, 139, 245 ff.
Full expression. Full sentence,
G: 187
Function, see Expressional /., Sen-
tential /.
Functional calculus,"^ I: 35; II: 96,
104
Functor, X3; I: 16, 54!., 72;
G: 188
Functor-expression, II: 84, 87;
G: 188
Functor- variable, II: 84; G: Z95
349
INDEX OF SUBJECTS
( 5 , 130 f.
General syntax, 153, 167
Genus, G: 170, 293
Heterological, 21 1, 218
“Higher, The”, 314 f.
G
Geometry, 7, 78 ff., 178, 229,
327
Gothic symbols, see Symbols
H
Homogeneous, 260
Hypothesis, 48, 318 fF.
•Identity, Sentence of, see Equation
Identity, Symbol of, I: 15, 19, 31,
49 ff.; II: 84; G: 243 ff.
Implication, I: 19 f., 32; II: 158;
G: 202 f., 235, 253 ff.
Implication, Strict, see ** Strict'*
“ Impredicable”, 138, 211
Impredicative, 162 ff.
In-, when not listed see the un-
prefixed word
Incompatible, I: 40; II: 117
Incompleteness of arithmetic, 173,
221 f.
Indefinable, 106, 114, 134, 218 f.,
221
Indefinite, ii; I: 45 f.; II. 99,
113, 160 ff., 165 if., 172; G:
198
Independent, II: 117; G: 174
Indices, I ; 17 f. ; II : 86
Junction, see Sentential junc-
tion
R, see Class of expressions
f, see Constant
I, see Logical
L-, G: i8of., i82ff., 262ff., 265 ff.
Language, i, 4; G: 167
Language, Symbolic, 3
Language-region, II: 88
Language, Science of, 9
Law of nature, 48, 52, 81, 148, 180,
185, 3071 3*8 ff., 321
Level, Level-number, II : 85 f. ;
G: 186 f., 261
Liar, Antinomy of the, 213, 214 f.,
217 f.
Indirect mode of speech, see Mode
Individual, G: 188, 195
Induction, Complete, I: 32 f., 38;
II : 92 f., 121
Incomplete, 317
“Inexpressible”, 282 f., 314
Infinite sentential class, 1 : 37, 39;
II: 100 ^
Intensional, 188, 242, 243 ff.
“Internal”, 304
Interpretation, 13 1, 132, 233, 239,
327
Intuitionism, 46 ff., 305
Inverted commas, 18, 155, 158 f.
Irreflexive, 234 f.
Irresoluble, 1 : 28; II: 94, 133 L;
G: 171, 221 f.
Isogenous, G: 169, 188, 274
Isolated, G: 170
Isomorphic, G : 224, 265
Junction symbols, I: 17, 18 f.;
II: 93; G: 201
K
‘K*, K-operator, -description, I:
16. 22 f., 30; II: 92, 146
L
Limit, Limited operator, I: 21;
G: 191
“ Logic”, I, 233, 257 ff., 278 ff.
Logic, Intensional, 256, 282
Logic of Modalities, see Modalities
Logical, I. 1 . I: 13, 14, 25, 38,
73; G: 177 f.
2. 1 . language, G: 178, 209 f.
3. 1. rules, G: 180 f.
4. 1. analysis, 7
Logicism, 300, 325 ff.
Logic of science, see Science
350
INDEX OF SUBJECTS
M&terial mode of speech, see Mode
Mathematics, see Arithmetic^ Num-
ber
Mathematics, Classical, 83, 98, 128,
I48» 230, 325
Meaning, 189, 288 ff.
** Meaningless”, 47, (82), 138, 162,
163, 283, 319, 321, 322
Metalogic, 9
Metamathematics, 9, 325 ff.
Metaphysics, 7 f., 278 f., 282-284,
309, 320
91, II : 84
Name, 12 f., 26, 189 f.
Name-language, 12, 189
Natural law, see Law
Nega^on, I: 19, 20; G: 202 f.
Negation, Double, 1 : 34 ; G : 202
Non-contradjctoriness, II .‘124,128 ;
G: 207 ff., 21 1
Non-contradictoriness, Proof of,
128, 134, 219
Non-denumerable, 221, 267 ff.
nu, see Zero symbol
Null, II: 134 f.; G: 262 f.
Object-language, 4, 160
Object-sentence, 277 f., 284
Open, I: 21, 66; G: 194
Operable, G: 192
Operand, I: 21; G: 192
Operator, 1 : 2i, 23 ; II : 83 f. ;
G: 191, 193
p, see Predicate-variable
P-, G: 180 ff., 184 f., 316
Perfect, G: 176
Phenomenology, 289, 305
“Philosophy”, 8, 52, 277-281, 332
Physicalism, 151, 320, 325
Physical language, 149 f., 178, 307,
315 ff., 322, 328 ff.
rules, 178, 180 f.
syntax, 57, 79 ff.
Position, Positional symbols, 12, 45
Postulate, see Axiomatic Method,
Primitive Sentence, Principle
pr. Predicate, 13 ; 1 : 16, 73 ; G: 188
^t. Predicate-expression, II: 83,
87, 134 ff.; G: 188, 191
Modalities, Logic of, G: 237, 246,
250-258, 303
Mode of speech, Autonymous, 238,
247 ff.
Formal, 239, 280 f., 288 ff.,
299 ff-'» 302 ff.
Indirect, 291 f.
Material, 237 f., 239, 286, 287 ff.,
297 ff., 302 ff., 308 ff.
Model, G : 272 f.
Molecular sentence, II: 88,
321
N
Null-content, 176
“Number”, 285, 293 f., 295, 300,
304 f > 31 1
Number, Cardinal, 139, 142 ff.,
326
Real, II : 147 ff- ; G : 207, 220, 305
Numerical expression, 1: 14, 26,72 ;
II: 87; G: 205
functor, G: 205
predicate, G: 205
symbol (Numeral), I: 14, 17, 24,
26, 59, 73; G: 20s
variable, 1 : 17 ; II : 84 ; G : 205
O
Operator, Descriptional, 22 ; G : 193
Limited, I: 21; G: 191
Sentential, I: 30; II: 92
Universal, 1 : 21 ; II : 93 ; G: 193,
196 f., 231
Ostension, ostensive definition, 80,
155
Predicate-variable, II: 84; G: 195
Premiss, 27
Premiss-class, G: 199
Principal expression, P-symbol,
G: 177
Principle of : Aussonderung, 98, 268
Complete Induction, 1 : 32 f.,
38; II: 92 f., 121
Comprehension, 98, 142
Contradiction, II: 125; G: 203
Double Negation, I: 34; II: 125
Excluded Middle, 1: 34,48; II :i25
Extensionality, II: 92, 98
Infinity, II: 81, 97, 140 f.
Limitation, 268 ff.
Reducibility, 86, 98, 142, 212
INDEX OF SUBJECTS 35 1
Selection, II: 92 f., 97, 121 Pseudo-problem, -sentence, 253,
Substitution with arguments, 278 f., 283 f., 289 fl., 304,
n 309 ff., 319, 322, 324,
Probability, 149, 307, 316 331
Proof, I: 29, 33 f., 75; II: 94!.; Psychologism, 26, 42, 260, 278,
G: 171 289
Protocol sentence, see Sentence Psychology, psychological language
Pseudo-object-sentence, "see Sen- 15 1, 315, 324
tence
Q
Q, see Correlation Quasi-syn tactical, G: 234, 236 ff..
Qualitative definition, 80 256 f., 285 ff.
Question, 296
R
Range, G: 199 f.
Realism, 301, 309
Real number, see Number
Reduction, II: 102 ff.
Reductum (‘ *)» H • *05
Reflexive, 261, 263
Refutable, I: 28; II: 94; G: 17 1.
275
Regressive definition, 23, 68 ;
11:88, 89
Regular sequence, 148 f.
0 , see Sentence
F, see Sentential variable
S-, 262 ff.
fa, see Sentential symbol
Schema, i . Of primitive sentences,
I: 29 f.; II: 91, 96
2. Of proofs and derivations,
I; 33 f.; II: 95 f.
Science, Logicof,7, 279-284, 33iff.
Semantics, 9, 249
Semasiology, sematology, 9
“Sense” (“Meaning”), 42, 184,
258 f., 285, 290
Sentence, 1 : 14, 25 f., 72; II: 88;
G: 169!., 252 f.
Sentence, Correlated syntactical,
234» 236
Elementary, 166; G: 195
Of identity, see Equation
Primitive, 29; I: 30, 74; II:
91 ff. ; G : 17 1. See Principle
Protocol, 305, 317 ff., 329 f.
Pseudo-object-, 234, 285 f.
Syntactical, 33 f., 284, 286 f.
Unlimited existential, 163
Related, G : 169 f.
Relation, 260, 262
Relativistic nature of lang^ge,
186, 245, 257> 299, 322
Replacement, 1 : 36 f. ; G : 169
Resoluble, i. r. S, ^ 1 , 46, 47;
II: 94, 113; G: 171
2. r. Language, G: 209
Resolution, Method of, 46, 47, 161
Rules of Inference, 27, 29; I: 32;
11:94
Sentential calculus, 1 : 30, 34 ;
II: 91 f., 96
class. Infinite, see Infinite
framework, G: 187, 191
function, 21, 137; G: 191
junction, G : 200 f.
operator, see Operator
symbol, I: 84; G: 169
variable, II: 84, 158; G: 195
Sequence of numbers, 148 f., 305
Series-number, 56 f., 60
0fu, see Sentential function
0g, see Sentential framework
Speech, see Mode
0t, see Accented expression
“Strict” implication, 203, 232,
237, 25IJ 254 ff-
0tu, see System of Levels
Sub-language, 179, 225
Substitution, I: 22, 32, 36 f., 73;
II: 90 ff., 96; G: 189, 191 f.,
193 f-
with arguments, 92 f., 125
Substitution-position, (-place), I:
22; G: 192
INDEX OF SUBJECTS
35 ^
Substitution-value, G: 191, 193
Symbol, 4; I; 16; G: 168
Symbolic language, 3
Symbols, Equal, 1 5
Gothic, I: 17 f.; II: 83; G:
169
Incomplete, 138
Numerical, see Numerical
Primitive, 1 : 23, 71, 77; II: 89;
G: 171, 231
Principal, G: 177
Sentential, see Sentential
Subsidiary, G ; 177
Symmetrical, G: 261, 263
Tautology, 44, 176; see Analytic
Term-number, 55, 60, 62, 68
Test (verification), 317 fF., 323
Tolerance, Principle of, 51 f., 164,
321
Total content, 176
Transform^nce, G : 223 IT.
Transformation rules, 2,4;! : 27 flf.,
73 ; II : 90 ff. ; G : 168 ff.
Un-, see the unprefixed word
Undefined, see Primitive symbols;
cf. Indefinable
Unity of science, language, 286 f.,
320 f.
SB, see Variable-expression
D, see Variable
Valid, G: 173 f.
Valuation, II: 107, 108
Value-expression, 1 : 83
Values, Course of, G: 187 f., 241
Range of, II: 90; G: 19 1
Variable, 1 : 16, 21 ; II : 84, 189 ff. ;
G: I94> 295
Variable, Numerical, see N.v.
Wahlfolge, 148, 305
3* see Numerical expression
3, see Numerical variable
55 f., 77
Zero-symbol, 1 : 13 f., 17 ; G : 205
Synonymous, i. s. I: 42; II:
120; G: 176, 184, 241
2. s. languages, G: 226
Syntactical correlation, 222
designation, 154!., 160
Syntax, l, 8
Syntax, Axiomatic, 79
Descriptive, see Descriptive
General, 153, 167
Physical, see Physical
Pure, 6, 15, 56 f., 78
Syntax-language, 4, 53, 153, 160;
G:235
Synthetic, I: 28, 40 f., loi, 115
T
Transitive, G: 261, 263
Translation, G : 224 ff. ; 228
Transposed, see Mode of Speech
“True**, 214, 236, 240
Truth-function, 240
Truth-value tables, 20, 201
Types, II: 84 ff., 98, 137 f-;
G: 164 f., 249, 298
Types, Rule of, 212
U
Universal, see Operator^ Variable
Universality, 21, 47 ff., 163 f., 198,
3*1
Universal word, 292, 310
Unordered sequence, 148 f.
V
Sentential, see S.v.
Universal, 165
Variable-expression, G: 191
Variability-number, 198
Variant, G: 193
Verification, see Test
DCrfn, Junction-symbols y I, II
; Vitalism, 319, 324
S3!, see Sentential junction
of, see Junction-symbols y G
w
Word-language, 2, 8, 227 f.
Z
3fu, see Numerical functor
3pr, see Numerical predicate
33, see Numerical symbols
Syktactical symbols (used in connection with Gothic symbols) :
{• •>» 34; +> 34; 0 (••)» 84; (; ;), see Substitution; [; :], see Replacement.
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Nature of Intelligence. By L. L. Thurstone.
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Primitive Mind and Modem Civilization. By C. R. Aldrich.
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Telepathy and Clairvoyance. By R. Tischner.
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Treatise on Induction and Probability. By G. H. von Wright.
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